69:
45:
260:
6425:
6180:
4974:
5464:
2774:
7331:
2149:
7244:
splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear
322:
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far
6170:
are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
1496:
3969:
1640:
4266:
of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.
1174:
3814:
4123:
2684:
4963:
4227:
3051:
7764:
7654:
323:
enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see
6005:
4775:
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at
3611:
3463:
2495:
2428:
6650:
1759:
6865:
5607:
7221:
4644:
7544:
5245:
1292:
7416:
5880:
6412:. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used although the term
7883:
1848:
1042:
974:
3346:
1360:
218:
Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as
3821:
1507:
5273:
If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as
4497:
7040:
2299:
2245:
786:
721:
6372:
4761:
247:, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of
3290:
3208:
413:
5142:
When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after
4374:
1205:(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or
1080:
3685:
7464:
5055:
2825:
2755:
1893:
304:
7312:
370:
7094:
6096:
877:
829:
6958:
6913:
4531:
3518:
depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
8027:
4020:
1963:
500:
5714:
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is
5137:
2575:
5146:
the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function
2112:
7973:
5756:
1319:
3516:
3493:
2078:
2055:
2541:
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function
2012:
4680:
1219:
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
7910:
1352:
5081:
4794:
4401:
4316:
2138:
1922:
617:-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as
4132:
7934:
6168:
2032:
1983:
1664:
544:
524:
473:
453:
433:
2895:
7668:
3356:
The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).
7558:
6460:
in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary
1216:
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
3620:
should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining
8296:
The
British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge
5895:
3535:
3394:
7245:
factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve
2433:
2366:
415:
where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of
8424:
8203:
6511:
5632:. From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote.
526:
become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of
546:, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the
5711:-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
5523:
8495:
4552:
1672:
7475:
5152:
1225:
7347:
435:
become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of
8391:
8365:
8279:
5788:
4544:
605:
asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as
7775:
1771:
1491:{\displaystyle \lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=+\infty ,}
980:
912:
3964:{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\left({\frac {2x^{2}+3x+1}{x}}-2x\right)=3}
3296:
1635:{\displaystyle \lim _{x\to 0^{-}}f(x)=\lim _{x\to 0^{-}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=-\infty }
6748:
8518:
7107:
6483:, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
8037:, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.
8075:
4418:
2250:
2196:
727:
662:
8070:
6304:
4694:
7096:, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a
3214:
8523:
8358:
3149:
375:
31:
7235:
splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals,
1169:{\displaystyle f(x)={\begin{cases}{\frac {1}{x}}&{\text{if }}x>0,\\5&{\text{if }}x\leq 0.\end{cases}}}
5388:
3809:{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {2x^{2}+3x+1}{x^{2}}}=2}
327:). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
7769:
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
6963:
6120:
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is
5143:
4322:
173:, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.
8503:
7427:
4980:
2780:
2709:
1853:
8123:
7913:
6665:
4263:
4259:
has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
1213:
at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
265:
193:
7259:
6476:
333:
7045:
6051:
842:
794:
307:
4118:{\displaystyle m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {\ln x}{x}}=0}
8227:
vol. 2, The
Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541
4503:
2679:{\displaystyle \lim _{x\to -\infty }{\frac {1}{x^{2}+1}}=\lim _{x\to +\infty }{\frac {1}{x^{2}+1}}=0.}
126:
is a line such that the distance between the curve and the line approaches zero as one or both of the
8195:
5277:
increases, but the quotient will not be linear, and the function does not have an oblique asymptote.
7978:
7223:
the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
1104:
478:
8223:
6461:
6433:
6014:
5086:
1210:
590:
559:
455:, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large
248:
236:) if the distance between the two curves tends to zero as they tend to infinity, although the term
189:
166:
139:
135:
8128:
8383:
8141:
6918:
6873:
1206:
8065:
7939:
5717:
1927:
1297:
8420:
8387:
8361:
8275:
8219:
8199:
6453:
4256:
3498:
3475:
2762:
2704:
2060:
2037:
147:
77:
8492:
8267:
8133:
8097:
7334:
Hyperbolas, obtained cutting the same right circular cone with a plane and their asymptotes.
6413:
5490:
4958:{\displaystyle f(x)={\frac {x^{2}-5x+6}{x^{3}-3x^{2}+2x}}={\frac {(x-2)(x-3)}{x(x-1)(x-2)}}}
4650:
87:
8434:
8401:
8246:
7888:
4222:{\displaystyle n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\ln x}
2083:
8499:
8430:
8397:
8242:
6449:
6441:
6409:
1986:
1328:
324:
210:, horizontal asymptotes are horizontal lines that the graph of the function approaches as
162:
8328:
8261:
5060:
4380:
4295:
2117:
1901:
7330:
3046:{\displaystyle \lim _{x\to +\infty }\left=0\,{\mbox{ or }}\lim _{x\to -\infty }\left=0.}
1992:
8241:, Translated from the first Russian ed. by L. F. Boron, Groningen: P. Noordhoff N. V.,
8183:
8046:
7919:
7759:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=0.}
6457:
6153:
6021:(a function of one real variable and returning real values). The graph of the function
2758:
2017:
1968:
1649:
529:
509:
458:
438:
418:
170:
154:
7649:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1}
68:
44:
8512:
8312:
6018:
558:-axis are asymptotes of the curve. These ideas are part of the basis of concept of a
259:
6445:
6424:
6179:
4973:
38:
8114:
Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the
Projective Plane",
5463:
372:
shown in this section. The coordinates of the points on the curve are of the form
8462:
8447:
5467:(sec(t), cosec(t)), or x + y = (xy), with 2 horizontal and 2 vertical asymptotes.
7550:
6429:
6000:{\displaystyle {\frac {|ax(\gamma (t))+by(\gamma (t))+c|}{\sqrt {a^{2}+b^{2}}}}}
17:
3606:{\displaystyle n\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}(f(x)-mx)}
8486:
8190:
Calculus, Vol. 1: One-Variable
Calculus with an Introduction to Linear Algebra
7104:, even when it does not have any parabola that is a curvilinear asymptote. If
2773:
2357:
2153:
8271:
8188:
7339:
3458:{\displaystyle m\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}f(x)/x}
2700:
6479:, as the intersection at infinity is of multiplicity at least two. For a
5695: → 0 from the right, and the distance between the curve and the
2689:
Other common functions that have one or two horizontal asymptotes include
2490:{\displaystyle \lim _{x\rightarrow +\infty }\arctan(x)={\frac {\pi }{2}}.}
2423:{\displaystyle \lim _{x\rightarrow -\infty }\arctan(x)=-{\frac {\pi }{2}}}
6397:
571:
2148:
609:
tends to +∞ or −∞. As the name indicates they are parallel to the
8482:
8145:
6645:{\displaystyle P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots +P_{1}(x,y)+P_{0}}
6255:
be another (unparameterized) curve. Suppose, as before, that the curve
143:
4271:
The cases of horizontal and oblique asymptotes for rational functions
8137:
902:–1), the numerator approaches 1 and the denominator approaches 0 as
613:-axis. Vertical asymptotes are vertical lines (perpendicular to the
6101:
This parameterization is to be considered over the open intervals (
562:
in mathematics, and this connection is explained more fully below.
7329:
6480:
6423:
6178:
5602:{\displaystyle \lim _{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty .}
5462:
4972:
2772:
2163:
are horizontal lines that the graph of the function approaches as
258:
123:
67:
43:
7772:
More generally, consider a surface that has an implicit equation
4639:{\displaystyle f(x)={\frac {2x^{2}+3x+5}{x}}=2x+3+{\frac {5}{x}}}
1754:{\displaystyle f'(x)={\frac {-(\cos({\tfrac {1}{x}})+1)}{x^{2}}}}
142:
and related contexts, an asymptote of a curve is a line which is
8030:
7539:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.}
5240:{\displaystyle f(x)={\frac {x^{2}+x+1}{x+1}}=x+{\frac {1}{x+1}}}
1287:{\displaystyle f(x)={\tfrac {1}{x}}+\sin({\tfrac {1}{x}})\quad }
165:+ σύν "together" + πτωτ-ός "fallen". The term was introduced by
7411:{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
6010:
which tends to zero simultaneously as the previous expression.
6672:. Vanishing of the linear factors of the highest degree term
6486:
A plane algebraic curve is defined by an equation of the form
5875:{\displaystyle {\frac {|ax(t)+by(t)+c|}{\sqrt {a^{2}+b^{2}}}}}
1074:
does not affect the asymptote. For example, for the function
56: = 0), and oblique asymptote (purple line, given by
6295:, when there is no risk of confusion with linear asymptotes.
5889:) is a change of parameterization then the distance becomes
5083:. Green: difference between the graph and its asymptote for
8493:
Hyperboloid and
Asymptotic Cone, string surface model, 1872
7878:{\displaystyle P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,}
1843:{\displaystyle x_{n}={\frac {(-1)^{n}}{(2n+1)\pi }},\quad }
1162:
1037:{\displaystyle \lim _{x\to 1^{-}}{\frac {x}{x-1}}=-\infty }
969:{\displaystyle \lim _{x\to 1^{+}}{\frac {x}{x-1}}=+\infty }
243:
Asymptotes convey information about the behavior of curves
111:
27:
Limit of the tangent line at a point that tends to infinity
3341:{\displaystyle =\lim _{x\to \pm \infty }{\frac {1}{x}}=0.}
8469:
This has a more general treatment of asymptotic surfaces.
570:
The asymptotes most commonly encountered in the study of
99:
93:
6860:{\displaystyle Q'_{x}(b,a)x+Q'_{y}(b,a)y+P_{n-1}(b,a)=0}
5517:)). Suppose that the curve tends to infinity, that is:
72:
A curve intersecting an asymptote infinitely many times.
7216:{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,}
5297:), then the translations of it also have an asymptote.
4269:
3360:
General computation of oblique asymptotes for functions
6033:) is the set of points of the plane with coordinates (
5671: → ∞ and the distance from the curve to the
2969:
2806:
1715:
1603:
1578:
1456:
1431:
1269:
1245:
285:
7981:
7942:
7922:
7891:
7778:
7671:
7561:
7478:
7430:
7350:
7318:
with multiplicity 4, leading to the unique asymptote
7314:, but its highest order term gives the linear factor
7262:
7110:
7048:
6966:
6921:
6876:
6751:
6514:
6307:
6156:
6054:
5898:
5791:
5720:
5526:
5155:
5089:
5063:
4983:
4797:
4697:
4653:
4555:
4506:
4421:
4383:
4325:
4298:
4135:
4023:
3824:
3688:
3538:
3501:
3478:
3397:
3299:
3217:
3152:
2898:
2783:
2712:
2578:
2436:
2369:
2253:
2199:
2120:
2086:
2063:
2040:
2020:
1995:
1971:
1930:
1904:
1856:
1774:
1675:
1652:
1510:
1363:
1331:
1300:
1228:
1209:) in more than one point. Moreover, if a function is
1083:
983:
915:
845:
797:
730:
665:
656:
if at least one of the following statements is true:
532:
512:
481:
461:
441:
421:
378:
336:
268:
240:
by itself is usually reserved for linear asymptotes.
96:
37:"Asymptotic" redirects here. Not to be confused with
8325:
The elementary differential geometry of plane curves
8263:
The elementary differential geometry of plane curves
4492:{\displaystyle f(x)={\frac {2x^{2}+7}{3x^{2}+x+12}}}
502:
is never 0, so the curve never actually touches the
114:
108:
105:
8102:
An elementary treatise on the differential calculus
2294:{\displaystyle \lim _{x\rightarrow +\infty }f(x)=c}
2240:{\displaystyle \lim _{x\rightarrow -\infty }f(x)=c}
781:{\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty ,}
716:{\displaystyle \lim _{x\to a^{-}}f(x)=\pm \infty ,}
102:
90:
8187:
8021:
7967:
7928:
7904:
7877:
7758:
7648:
7538:
7458:
7410:
7306:
7215:
7088:
7034:
6952:
6907:
6859:
6644:
6366:
6162:
6090:
5999:
5874:
5750:
5601:
5239:
5131:
5075:
5049:
4957:
4755:
4674:
4638:
4525:
4491:
4395:
4368:
4310:
4221:
4117:
3963:
3808:
3605:
3510:
3487:
3457:
3340:
3284:
3202:
3045:
2819:
2749:
2678:
2530:is a horizontal asymptote for the arctangent when
2511:is a horizontal asymptote for the arctangent when
2489:
2422:
2293:
2239:
2132:
2106:
2072:
2049:
2026:
2006:
1977:
1957:
1924:both from the left and from the right, the values
1916:
1887:
1842:
1753:
1658:
1634:
1490:
1346:
1313:
1286:
1168:
1036:
968:
871:
823:
780:
715:
538:
518:
494:
467:
447:
427:
407:
364:
298:
8033:which is centered at the origin. It is called an
7469:The equation for the union of these two lines is
5635:For example, the upper right branch of the curve
6367:{\displaystyle y={\frac {x^{3}+2x^{2}+3x+4}{x}}}
5528:
4192:
4143:
4073:
4031:
3881:
3832:
3738:
3696:
3561:
3420:
3304:
3222:
3154:
2976:
2900:
2627:
2580:
2438:
2371:
2255:
2201:
1550:
1512:
1403:
1365:
985:
917:
847:
799:
732:
667:
5455:=0+2=2, and no vertical or oblique asymptotes.
5387:If a known function has an asymptote, then the
4756:{\displaystyle f(x)={\frac {2x^{4}}{3x^{2}+1}}}
2847:When a linear asymptote is not parallel to the
6436:(solid) with a single real asymptote (dashed).
5285:If a known function has an asymptote (such as
3285:{\displaystyle =\lim _{x\to \pm \infty }\left}
6681:defines the asymptotes of the curve: setting
3352:Elementary methods for identifying asymptotes
3203:{\displaystyle \lim _{x\to \pm \infty }\left}
3082:tends to +∞, and in the second case the line
408:{\displaystyle \left(x,{\frac {1}{x}}\right)}
161:) which means "not falling together", from ἀ
8:
8327:Cambridge, University Press, 1920, pp 89ff.(
6227:be a parametric plane curve, in coordinates
6013:An important case is when the curve is the
4766:
48:The graph of a function with a horizontal (
5687:-axis is an asymptote of the curve. Also,
3542:
3401:
2178:is a horizontal asymptote of the function
1325:This function has a vertical asymptote at
8127:
7986:
7980:
7947:
7941:
7921:
7896:
7890:
7860:
7817:
7783:
7777:
7742:
7732:
7726:
7715:
7705:
7699:
7688:
7678:
7672:
7670:
7632:
7622:
7616:
7605:
7595:
7589:
7578:
7568:
7562:
7560:
7522:
7512:
7506:
7495:
7485:
7479:
7477:
7440:
7429:
7394:
7384:
7378:
7367:
7357:
7351:
7349:
7293:
7285:
7271:
7263:
7261:
7177:
7146:
7115:
7109:
7053:
7047:
7035:{\displaystyle Q'_{x}(b,a)=Q'_{y}(b,a)=0}
7002:
6971:
6965:
6926:
6920:
6881:
6875:
6824:
6790:
6756:
6750:
6636:
6608:
6568:
6540:
6513:
6337:
6321:
6314:
6306:
6291:is simply referred to as an asymptote of
6155:
6053:
5988:
5975:
5964:
5902:
5899:
5897:
5863:
5850:
5839:
5795:
5792:
5790:
5719:
5572:
5550:
5531:
5525:
5219:
5178:
5171:
5154:
5088:
5062:
5027:
5006:
4982:
4884:
4863:
4847:
4820:
4813:
4796:
4738:
4723:
4713:
4696:
4652:
4626:
4581:
4571:
4554:
4513:
4505:
4468:
4447:
4437:
4420:
4382:
4369:{\displaystyle f(x)={\frac {1}{x^{2}+1}}}
4351:
4341:
4324:
4297:
4195:
4146:
4134:
4091:
4076:
4061:
4034:
4022:
3914:
3904:
3884:
3835:
3823:
3792:
3766:
3756:
3741:
3726:
3699:
3687:
3564:
3559:
3551:
3546:
3544:
3543:
3537:
3500:
3477:
3447:
3423:
3418:
3410:
3405:
3403:
3402:
3396:
3322:
3307:
3298:
3256:
3225:
3216:
3157:
3151:
2979:
2968:
2967:
2903:
2897:
2805:
2782:
2735:
2711:
2655:
2645:
2630:
2608:
2598:
2583:
2577:
2474:
2441:
2435:
2410:
2374:
2368:
2258:
2252:
2204:
2198:
2119:
2085:
2062:
2039:
2019:
1994:
1970:
1946:
1929:
1903:
1855:
1804:
1788:
1779:
1773:
1743:
1714:
1696:
1674:
1651:
1602:
1577:
1564:
1553:
1526:
1515:
1509:
1455:
1430:
1417:
1406:
1379:
1368:
1362:
1330:
1299:
1268:
1244:
1227:
1145:
1119:
1107:
1099:
1082:
1007:
999:
988:
982:
939:
931:
920:
914:
861:
850:
844:
813:
802:
796:
746:
735:
729:
681:
670:
664:
531:
511:
482:
480:
460:
440:
420:
390:
377:
352:
335:
284:
267:
8448:L.P. Siceloff, G. Wentworth, D.E. Smith
6267:if the shortest distance from the point
5391:of the function also have an asymptote.
4969:Oblique asymptotes of rational functions
3364:The oblique asymptote, for the function
2147:
839:from the left (from lesser values), and
8355:Elementary Geometry of Algebraic Curves
8342:An elementary treatise on curve tracing
8309:An elementary treatise on curve tracing
8088:
1047:and the curve has a vertical asymptote
153:The word asymptote is derived from the
8417:Introduction to plane algebraic curves
7459:{\displaystyle y=\pm {\frac {b}{a}}x.}
7256:has no real points outside the square
6448:are the lines that are tangent to the
5266:. This is because the other term, 1/(
5050:{\displaystyle f(x)=(x^{2}+x+1)/(x+1)}
3143: = 0) as seen in the limits
2820:{\displaystyle f(x)=x+{\tfrac {1}{x}}}
2750:{\displaystyle x\mapsto \exp(-x^{2}),}
6408:Asymptotes are used in procedures of
6132:. The non-vertical case has equation
2871:) is asymptotic to the straight line
2156:function has two different asymptotes
2114:doesn't have a vertical asymptote at
1888:{\displaystyle \quad n=0,1,2,\ldots }
1066:, and its precise value at the point
176:There are three kinds of asymptotes:
7:
8380:A treatise on algebraic plane curves
6471:intersects its asymptote at most at
6456:. For example, one may identify the
6045:)). For this, a parameterization is
5250:shown to the right. As the value of
4547:of the numerator by the denominator
4413:= the ratio of leading coefficients
299:{\displaystyle f(x)={\tfrac {1}{x}}}
7307:{\displaystyle |x|\leq 1,|y|\leq 1}
506:-axis. Similarly, as the values of
365:{\displaystyle f(x)={\frac {1}{x}}}
330:Consider the graph of the function
7089:{\displaystyle P_{n-1}(b,a)\neq 0}
6091:{\displaystyle t\mapsto (t,f(t)).}
5593:
5281:Transformations of known functions
4205:
4156:
4086:
4044:
3894:
3845:
3751:
3709:
3505:
3482:
3388:is computed first and is given by
3317:
3235:
3167:
3127:, which has the oblique asymptote
2989:
2913:
2640:
2593:
2451:
2384:
2268:
2214:
2067:
2044:
1629:
1482:
1031:
963:
872:{\displaystyle \lim _{x\to a^{+}}}
824:{\displaystyle \lim _{x\to a^{-}}}
772:
707:
25:
8419:, Boston, MA: Birkhäuser Boston,
5758:then the distance from the point
5643:can be defined parametrically as
4251:Asymptotes for rational functions
3372:), will be given by the equation
8378:Coolidge, Julian Lowell (1959),
6458:asymptotes to the unit hyperbola
6190:+3 is a parabolic asymptote to (
4526:{\displaystyle y={\frac {2}{3}}}
4276:deg(numerator)−deg(denominator)
4243:does not have an asymptote when
86:
8266:, Cambridge, University Press,
5683: → ∞. Therefore, the
5616:if the distance from the point
1857:
1839:
1301:
1283:
1062:) may or may not be defined at
228:More generally, one curve is a
8022:{\displaystyle P_{d}(x,y,z)=0}
8010:
7992:
7847:
7829:
7807:
7789:
7294:
7286:
7272:
7264:
7201:
7189:
7167:
7155:
7136:
7124:
7077:
7065:
7023:
7011:
6992:
6980:
6947:
6935:
6902:
6890:
6848:
6836:
6811:
6799:
6777:
6765:
6626:
6614:
6592:
6580:
6558:
6546:
6530:
6518:
6404:Asymptotes and curve sketching
6263:is a curvilinear asymptote of
6082:
6079:
6073:
6061:
6058:
5965:
5955:
5952:
5946:
5940:
5928:
5925:
5919:
5913:
5903:
5840:
5830:
5824:
5812:
5806:
5796:
5587:
5584:
5578:
5562:
5556:
5543:
5535:
5165:
5159:
5044:
5032:
5024:
4999:
4993:
4987:
4949:
4937:
4934:
4922:
4914:
4902:
4899:
4887:
4807:
4801:
4707:
4701:
4565:
4559:
4431:
4425:
4335:
4329:
4199:
4185:
4173:
4167:
4161:
4150:
4080:
4058:
4052:
4038:
3888:
3874:
3862:
3856:
3850:
3839:
3745:
3723:
3717:
3703:
3600:
3588:
3582:
3576:
3568:
3444:
3438:
3427:
3311:
3229:
3186:
3180:
3161:
3029:
3014:
3008:
3002:
2983:
2953:
2938:
2932:
2926:
2907:
2793:
2787:
2741:
2725:
2716:
2634:
2587:
2553:has a horizontal asymptote at
2468:
2462:
2445:
2401:
2395:
2378:
2282:
2276:
2262:
2228:
2222:
2208:
2101:
2095:
1952:
1939:
1827:
1812:
1801:
1791:
1735:
1726:
1711:
1702:
1690:
1684:
1557:
1543:
1537:
1519:
1410:
1396:
1390:
1372:
1280:
1265:
1238:
1232:
1194:) has the vertical asymptote
1093:
1087:
992:
924:
854:
806:
763:
757:
739:
698:
692:
674:
589:. These can be computed using
495:{\displaystyle {\frac {1}{x}}}
346:
340:
278:
272:
1:
8171:History of Mathematics, vol 2
8096:Williamson, Benjamin (1899),
6400:rather than a straight line.
6259:tends to infinity. The curve
5372:is a horizontal asymptote of
5352:is a horizontal asymptote of
5132:{\displaystyle x=1,2,3,4,5,6}
641:of the graph of the function
6377:has a curvilinear asymptote
5612:A line ℓ is an asymptote of
5493:plane curve, in coordinates
5451:+2 has horizontal asymptote
3638:is the oblique asymptote of
232:of another (as opposed to a
8071:Encyclopedia of Mathematics
6953:{\displaystyle Q'_{y}(b,a)}
6908:{\displaystyle Q'_{x}(b,a)}
5782:)) to the line is given by
5329:is a vertical asymptote of
5309:is a vertical asymptote of
4765:no linear asymptote, but a
3096:is an oblique asymptote of
3070:is an oblique asymptote of
3056:In the first case the line
1765:For the sequence of points
224:+∞ or −∞.
216:+∞ or −∞.
52: = 0), vertical (
8540:
8359:Cambridge University Press
8194:(2nd ed.), New York:
7226:Over the complex numbers,
6502:is a polynomial of degree
6475:−2 other points, by
6298:For example, the function
3657:For example, the function
1207:a vertical line in general
574:are of curves of the form
188:. For curves given by the
36:
32:Asymptote (disambiguation)
29:
8237:Pogorelov, A. V. (1959),
8158:Oxford English Dictionary
7968:{\displaystyle P_{d-1}=0}
5751:{\displaystyle ax+by+c=0}
5258:approaches the asymptote
1958:{\displaystyle f'(x_{n})}
1314:{\displaystyle \quad x=0}
621:tends to +∞ or −∞.
318:-axis are the asymptotes.
6467:A plane curve of degree
5624:) to ℓ tends to zero as
3511:{\displaystyle +\infty }
3488:{\displaystyle -\infty }
2073:{\displaystyle -\infty }
2050:{\displaystyle +\infty }
475:becomes, its reciprocal
8218:Reference for section:
8160:, second edition, 1989.
8064:Kuptsov, L.P. (2001) ,
7914:homogeneous polynomials
7421:has the two asymptotes
6960:are not both zero. If
6434:the folium of Descartes
6416:seems to be preferred.
6128:, for some real number
5707: → 0. So the
4229:, which does not exist.
2855:-axis, it is called an
2569:because, respectively,
2170:. The horizontal line
566:Asymptotes of functions
8260:Fowler, R. H. (1920),
8023:
7969:
7930:
7906:
7879:
7760:
7650:
7540:
7460:
7412:
7335:
7308:
7217:
7090:
7036:
6954:
6909:
6861:
6646:
6498:) = 0 where
6437:
6392:, which is known as a
6368:
6206:
6175:Curvilinear asymptotes
6164:
6092:
6001:
5876:
5752:
5703:which approaches 0 as
5679:which approaches 0 as
5603:
5468:
5241:
5139:
5133:
5077:
5051:
4959:
4757:
4676:
4675:{\displaystyle y=2x+3}
4640:
4543:= the quotient of the
4527:
4493:
4397:
4370:
4312:
4285:Asymptote for example
4279:Asymptotes in general
4223:
4119:
3965:
3810:
3607:
3512:
3489:
3459:
3342:
3286:
3204:
3047:
2844:
2821:
2751:
2680:
2491:
2424:
2295:
2241:
2157:
2134:
2108:
2074:
2051:
2028:
2008:
1979:
1959:
1918:
1889:
1844:
1755:
1660:
1636:
1492:
1348:
1315:
1288:
1170:
1038:
970:
873:
825:
782:
717:
550:-axis. Thus, both the
540:
520:
496:
469:
449:
429:
409:
366:
319:
300:
73:
65:
8519:Mathematical analysis
8357:, § 12.6 Asymptotes,
8329:online at archive.org
8239:Differential geometry
8196:John Wiley & Sons
8024:
7970:
7931:
7907:
7905:{\displaystyle P_{i}}
7880:
7761:
7651:
7541:
7461:
7413:
7333:
7309:
7218:
7091:
7037:
6955:
6910:
6862:
6647:
6440:The asymptotes of an
6427:
6369:
6182:
6165:
6093:
6002:
5877:
5753:
5604:
5466:
5242:
5134:
5078:
5057:. Red: the asymptote
5052:
4976:
4960:
4767:curvilinear asymptote
4758:
4677:
4641:
4528:
4494:
4398:
4371:
4313:
4224:
4120:
3966:
3811:
3608:
3513:
3490:
3460:
3343:
3287:
3205:
3048:
2822:
2776:
2752:
2681:
2492:
2425:
2296:
2242:
2161:Horizontal asymptotes
2151:
2144:Horizontal asymptotes
2135:
2109:
2107:{\displaystyle f'(x)}
2075:
2052:
2029:
2009:
1980:
1960:
1919:
1890:
1845:
1756:
1661:
1637:
1493:
1349:
1316:
1289:
1179:has a limit of +∞ as
1171:
1039:
971:
874:
835:approaches the value
826:
783:
718:
541:
521:
497:
470:
450:
430:
410:
367:
308:Cartesian coordinates
301:
262:
230:curvilinear asymptote
71:
47:
8415:Kunz, Ernst (2005),
8224:The Penny Cyclopædia
8116:Mathematics Magazine
7979:
7975:. Then the equation
7940:
7920:
7889:
7776:
7669:
7659:is said to have the
7559:
7476:
7428:
7348:
7260:
7108:
7046:
6964:
6919:
6874:
6749:
6512:
6305:
6154:
6113:can be −∞ and
6052:
5896:
5789:
5718:
5524:
5153:
5087:
5061:
4981:
4977:Black: the graph of
4795:
4695:
4651:
4553:
4504:
4419:
4381:
4323:
4296:
4133:
4021:
3985:is the asymptote of
3822:
3686:
3536:
3499:
3476:
3395:
3297:
3215:
3150:
2896:
2843:are both asymptotes.
2781:
2710:
2576:
2557: = 0 when
2434:
2367:
2329:, and in the second
2251:
2197:
2118:
2084:
2061:
2038:
2018:
1993:
1969:
1928:
1902:
1854:
1772:
1673:
1650:
1508:
1361:
1347:{\displaystyle x=0,}
1329:
1298:
1226:
1081:
981:
913:
843:
795:
728:
663:
593:and classified into
530:
510:
479:
459:
439:
419:
376:
334:
266:
30:For other uses, see
8353:C.G. Gibson (1998)
8294:William Nicholson,
8173:Dover (1958) p. 318
8083:Specific references
7154:
7123:
7010:
6979:
6934:
6889:
6870:is an asymptote if
6798:
6764:
6450:projectivized curve
6394:parabolic asymptote
5427:is an asymptote of
5407:is an asymptote of
5270:+1), approaches 0.
5076:{\displaystyle y=x}
4396:{\displaystyle y=0}
4311:{\displaystyle y=0}
4272:
3529:can be computed by
3525:then the value for
3108:tends to −∞.
2889: ≠ 0) if
2360:function satisfies
2345:as an asymptote as
2305:In the first case,
2133:{\displaystyle x=0}
1917:{\displaystyle x=0}
625:Vertical asymptotes
249:asymptotic analysis
167:Apollonius of Perga
140:projective geometry
8498:2012-02-15 at the
8405:, pp. 40–44.
8384:Dover Publications
8058:General references
8019:
7965:
7926:
7902:
7875:
7756:
7646:
7536:
7456:
7408:
7336:
7304:
7213:
7142:
7111:
7086:
7032:
6998:
6967:
6950:
6922:
6905:
6877:
6857:
6786:
6752:
6642:
6438:
6364:
6207:
6160:
6088:
5997:
5872:
5748:
5691: → ∞ as
5667: → ∞ as
5599:
5542:
5469:
5459:General definition
5237:
5140:
5129:
5073:
5047:
4955:
4753:
4672:
4636:
4545:Euclidean division
4523:
4489:
4393:
4366:
4308:
4270:
4219:
4209:
4160:
4115:
4090:
4048:
3961:
3898:
3849:
3806:
3755:
3713:
3603:
3575:
3508:
3485:
3455:
3434:
3338:
3321:
3282:
3239:
3200:
3171:
3043:
2993:
2973:
2917:
2845:
2835:= 0) and the line
2817:
2815:
2769:Oblique asymptotes
2747:
2703:as it graph), the
2676:
2644:
2597:
2487:
2455:
2420:
2388:
2321:as asymptote when
2291:
2272:
2237:
2218:
2158:
2130:
2104:
2070:
2047:
2024:
2007:{\displaystyle f'}
2004:
1985:. Therefore, both
1975:
1955:
1914:
1885:
1840:
1751:
1724:
1656:
1646:The derivative of
1632:
1612:
1587:
1571:
1533:
1488:
1465:
1440:
1424:
1386:
1344:
1311:
1284:
1278:
1254:
1166:
1161:
1034:
1006:
966:
938:
890:For example, if ƒ(
869:
868:
821:
820:
778:
753:
713:
688:
639:vertical asymptote
536:
516:
492:
465:
445:
425:
405:
362:
320:
296:
294:
146:to the curve at a
74:
66:
8524:Analytic geometry
8450:Analytic geometry
8426:978-0-8176-4381-2
8272:2027/uc1.b4073882
8205:978-0-471-00005-1
7929:{\displaystyle i}
7748:
7721:
7694:
7638:
7611:
7584:
7528:
7501:
7448:
7400:
7373:
6454:point at infinity
6362:
6279:tends to zero as
6163:{\displaystyle n}
5995:
5994:
5870:
5869:
5527:
5235:
5208:
4953:
4879:
4773:
4772:
4751:
4634:
4606:
4521:
4487:
4364:
4257:rational function
4191:
4142:
4107:
4072:
4030:
3939:
3880:
3831:
3798:
3737:
3695:
3624:exist. Otherwise
3560:
3556:
3554:
3419:
3415:
3413:
3330:
3303:
3264:
3221:
3153:
2975:
2972:
2899:
2857:oblique asymptote
2814:
2763:logistic function
2705:Gaussian function
2668:
2626:
2621:
2579:
2482:
2437:
2418:
2370:
2356:For example, the
2254:
2200:
2027:{\displaystyle 0}
1978:{\displaystyle 0}
1898:that approaches
1834:
1749:
1723:
1659:{\displaystyle f}
1611:
1586:
1549:
1511:
1464:
1439:
1402:
1364:
1277:
1253:
1148:
1122:
1115:
1023:
984:
955:
916:
906:approaches 1. So
846:
798:
731:
666:
539:{\displaystyle y}
519:{\displaystyle x}
490:
468:{\displaystyle x}
448:{\displaystyle y}
428:{\displaystyle x}
398:
360:
293:
148:point at infinity
136:tends to infinity
78:analytic geometry
16:(Redirected from
8531:
8470:
8460:
8454:
8445:
8439:
8437:
8412:
8406:
8404:
8375:
8369:
8351:
8345:
8338:
8332:
8321:
8315:
8305:
8299:
8292:
8286:
8284:
8257:
8251:
8249:
8234:
8228:
8216:
8210:
8208:
8193:
8180:
8174:
8167:
8161:
8155:
8149:
8148:
8131:
8111:
8105:
8104:
8093:
8078:
8028:
8026:
8025:
8020:
7991:
7990:
7974:
7972:
7971:
7966:
7958:
7957:
7935:
7933:
7932:
7927:
7911:
7909:
7908:
7903:
7901:
7900:
7884:
7882:
7881:
7876:
7865:
7864:
7828:
7827:
7788:
7787:
7765:
7763:
7762:
7757:
7749:
7747:
7746:
7737:
7736:
7727:
7722:
7720:
7719:
7710:
7709:
7700:
7695:
7693:
7692:
7683:
7682:
7673:
7655:
7653:
7652:
7647:
7639:
7637:
7636:
7627:
7626:
7617:
7612:
7610:
7609:
7600:
7599:
7590:
7585:
7583:
7582:
7573:
7572:
7563:
7545:
7543:
7542:
7537:
7529:
7527:
7526:
7517:
7516:
7507:
7502:
7500:
7499:
7490:
7489:
7480:
7465:
7463:
7462:
7457:
7449:
7441:
7417:
7415:
7414:
7409:
7401:
7399:
7398:
7389:
7388:
7379:
7374:
7372:
7371:
7362:
7361:
7352:
7313:
7311:
7310:
7305:
7297:
7289:
7275:
7267:
7255:
7222:
7220:
7219:
7214:
7188:
7187:
7150:
7119:
7102:
7101:
7100:parabolic branch
7095:
7093:
7092:
7087:
7064:
7063:
7041:
7039:
7038:
7033:
7006:
6975:
6959:
6957:
6956:
6951:
6930:
6914:
6912:
6911:
6906:
6885:
6866:
6864:
6863:
6858:
6835:
6834:
6794:
6760:
6742:, then the line
6741:
6695:
6651:
6649:
6648:
6643:
6641:
6640:
6613:
6612:
6579:
6578:
6545:
6544:
6477:Bézout's theorem
6420:Algebraic curves
6414:asymptotic curve
6396:because it is a
6391:
6373:
6371:
6370:
6365:
6363:
6358:
6342:
6341:
6326:
6325:
6315:
6275:) to a point on
6226:
6169:
6167:
6166:
6161:
6145:
6097:
6095:
6094:
6089:
6006:
6004:
6003:
5998:
5996:
5993:
5992:
5980:
5979:
5970:
5969:
5968:
5906:
5900:
5881:
5879:
5878:
5873:
5871:
5868:
5867:
5855:
5854:
5845:
5844:
5843:
5799:
5793:
5757:
5755:
5754:
5749:
5663:> 0). First,
5608:
5606:
5605:
5600:
5577:
5576:
5555:
5554:
5541:
5488:
5246:
5244:
5243:
5238:
5236:
5234:
5220:
5209:
5207:
5196:
5183:
5182:
5172:
5138:
5136:
5135:
5130:
5082:
5080:
5079:
5074:
5056:
5054:
5053:
5048:
5031:
5011:
5010:
4964:
4962:
4961:
4956:
4954:
4952:
4917:
4885:
4880:
4878:
4868:
4867:
4852:
4851:
4841:
4825:
4824:
4814:
4784:= 1, but not at
4762:
4760:
4759:
4754:
4752:
4750:
4743:
4742:
4729:
4728:
4727:
4714:
4681:
4679:
4678:
4673:
4645:
4643:
4642:
4637:
4635:
4627:
4607:
4602:
4586:
4585:
4572:
4532:
4530:
4529:
4524:
4522:
4514:
4498:
4496:
4495:
4490:
4488:
4486:
4473:
4472:
4459:
4452:
4451:
4438:
4402:
4400:
4399:
4394:
4375:
4373:
4372:
4367:
4365:
4363:
4356:
4355:
4342:
4317:
4315:
4314:
4309:
4273:
4242:
4228:
4226:
4225:
4220:
4208:
4159:
4124:
4122:
4121:
4116:
4108:
4103:
4092:
4089:
4065:
4047:
4013:
3984:
3970:
3968:
3967:
3962:
3954:
3950:
3940:
3935:
3919:
3918:
3905:
3897:
3848:
3815:
3813:
3812:
3807:
3799:
3797:
3796:
3787:
3771:
3770:
3757:
3754:
3730:
3712:
3678:
3637:
3612:
3610:
3609:
3604:
3574:
3558:
3557:
3555:
3552:
3550:
3545:
3517:
3515:
3514:
3509:
3494:
3492:
3491:
3486:
3464:
3462:
3461:
3456:
3451:
3433:
3417:
3416:
3414:
3411:
3409:
3404:
3384:. The value for
3347:
3345:
3344:
3339:
3331:
3323:
3320:
3291:
3289:
3288:
3283:
3281:
3277:
3270:
3266:
3265:
3257:
3238:
3209:
3207:
3206:
3201:
3199:
3195:
3170:
3139: = 1,
3095:
3069:
3052:
3050:
3049:
3044:
3036:
3032:
2992:
2974:
2970:
2960:
2956:
2916:
2884:
2826:
2824:
2823:
2818:
2816:
2807:
2777:In the graph of
2756:
2754:
2753:
2748:
2740:
2739:
2698:
2685:
2683:
2682:
2677:
2669:
2667:
2660:
2659:
2646:
2643:
2622:
2620:
2613:
2612:
2599:
2596:
2568:
2564:
2552:
2537:
2529:
2527:
2518:
2510:
2508:
2496:
2494:
2493:
2488:
2483:
2475:
2454:
2429:
2427:
2426:
2421:
2419:
2411:
2387:
2352:
2328:
2300:
2298:
2297:
2292:
2271:
2246:
2244:
2243:
2238:
2217:
2169:
2168:→ ±∞
2139:
2137:
2136:
2131:
2113:
2111:
2110:
2105:
2094:
2079:
2077:
2076:
2071:
2056:
2054:
2053:
2048:
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
2003:
1987:one-sided limits
1984:
1982:
1981:
1976:
1964:
1962:
1961:
1956:
1951:
1950:
1938:
1923:
1921:
1920:
1915:
1894:
1892:
1891:
1886:
1849:
1847:
1846:
1841:
1835:
1833:
1810:
1809:
1808:
1789:
1784:
1783:
1760:
1758:
1757:
1752:
1750:
1748:
1747:
1738:
1725:
1716:
1697:
1683:
1666:is the function
1665:
1663:
1662:
1657:
1641:
1639:
1638:
1633:
1622:
1618:
1617:
1613:
1604:
1588:
1579:
1570:
1569:
1568:
1532:
1531:
1530:
1497:
1495:
1494:
1489:
1475:
1471:
1470:
1466:
1457:
1441:
1432:
1423:
1422:
1421:
1385:
1384:
1383:
1353:
1351:
1350:
1345:
1320:
1318:
1317:
1312:
1293:
1291:
1290:
1285:
1279:
1270:
1255:
1246:
1200:
1185:
1175:
1173:
1172:
1167:
1165:
1164:
1149:
1146:
1123:
1120:
1116:
1108:
1043:
1041:
1040:
1035:
1024:
1022:
1008:
1005:
1004:
1003:
975:
973:
972:
967:
956:
954:
940:
937:
936:
935:
887:from the right.
879:is the limit as
878:
876:
875:
870:
867:
866:
865:
831:is the limit as
830:
828:
827:
822:
819:
818:
817:
787:
785:
784:
779:
752:
751:
750:
722:
720:
719:
714:
687:
686:
685:
655:
588:
545:
543:
542:
537:
525:
523:
522:
517:
501:
499:
498:
493:
491:
483:
474:
472:
471:
466:
454:
452:
451:
446:
434:
432:
431:
426:
414:
412:
411:
406:
404:
400:
399:
391:
371:
369:
368:
363:
361:
353:
305:
303:
302:
297:
295:
286:
234:linear asymptote
225:
217:
209:
121:
120:
117:
116:
113:
110:
107:
104:
101:
98:
95:
92:
21:
18:Parabolic branch
8539:
8538:
8534:
8533:
8532:
8530:
8529:
8528:
8509:
8508:
8500:Wayback Machine
8479:
8474:
8473:
8461:
8457:
8446:
8442:
8427:
8414:
8413:
8409:
8394:
8377:
8376:
8372:
8352:
8348:
8339:
8335:
8322:
8318:
8306:
8302:
8293:
8289:
8282:
8259:
8258:
8254:
8236:
8235:
8231:
8217:
8213:
8206:
8184:Apostol, Tom M.
8182:
8181:
8177:
8168:
8164:
8156:
8152:
8138:10.2307/2690881
8113:
8112:
8108:
8095:
8094:
8090:
8063:
8055:
8043:
8035:asymptotic cone
7982:
7977:
7976:
7943:
7938:
7937:
7918:
7917:
7892:
7887:
7886:
7856:
7813:
7779:
7774:
7773:
7738:
7728:
7711:
7701:
7684:
7674:
7667:
7666:
7661:asymptotic cone
7628:
7618:
7601:
7591:
7574:
7564:
7557:
7556:
7549:Similarly, the
7518:
7508:
7491:
7481:
7474:
7473:
7426:
7425:
7390:
7380:
7363:
7353:
7346:
7345:
7328:
7326:Asymptotic cone
7258:
7257:
7246:
7243:
7234:
7173:
7106:
7105:
7099:
7098:
7049:
7044:
7043:
6962:
6961:
6917:
6916:
6872:
6871:
6820:
6747:
6746:
6731:
6705:
6697:
6694:
6682:
6680:
6663:
6632:
6604:
6564:
6536:
6510:
6509:
6442:algebraic curve
6422:
6410:curve sketching
6406:
6378:
6333:
6317:
6316:
6303:
6302:
6235:) = (
6210:
6177:
6152:
6151:
6133:
6050:
6049:
5984:
5971:
5901:
5894:
5893:
5859:
5846:
5794:
5787:
5786:
5766:) = (
5716:
5715:
5655: = 1/
5639: = 1/
5568:
5546:
5522:
5521:
5501:) = (
5472:
5461:
5283:
5224:
5197:
5174:
5173:
5151:
5150:
5085:
5084:
5059:
5058:
5002:
4979:
4978:
4971:
4918:
4886:
4859:
4843:
4842:
4816:
4815:
4793:
4792:
4734:
4730:
4719:
4715:
4693:
4692:
4649:
4648:
4577:
4573:
4551:
4550:
4502:
4501:
4464:
4460:
4443:
4439:
4417:
4416:
4379:
4378:
4347:
4346:
4321:
4320:
4294:
4293:
4253:
4234:
4131:
4130:
4093:
4019:
4018:
4001:
3975:
3910:
3906:
3903:
3899:
3820:
3819:
3788:
3762:
3758:
3684:
3683:
3658:
3625:
3534:
3533:
3497:
3496:
3474:
3473:
3393:
3392:
3362:
3354:
3295:
3294:
3249:
3245:
3244:
3240:
3213:
3212:
3176:
3172:
3148:
3147:
3083:
3057:
2998:
2994:
2922:
2918:
2894:
2893:
2872:
2861:slant asymptote
2779:
2778:
2771:
2731:
2708:
2707:
2690:
2651:
2650:
2604:
2603:
2574:
2573:
2566:
2562:
2542:
2535:
2525:
2520:
2516:
2506:
2501:
2432:
2431:
2365:
2364:
2350:
2326:
2249:
2248:
2195:
2194:
2164:
2146:
2116:
2115:
2087:
2082:
2081:
2059:
2058:
2036:
2035:
2034:can be neither
2016:
2015:
1996:
1991:
1990:
1967:
1966:
1965:are constantly
1942:
1931:
1926:
1925:
1900:
1899:
1852:
1851:
1811:
1800:
1790:
1775:
1770:
1769:
1739:
1698:
1676:
1671:
1670:
1648:
1647:
1598:
1576:
1572:
1560:
1522:
1506:
1505:
1451:
1429:
1425:
1413:
1375:
1359:
1358:
1327:
1326:
1296:
1295:
1224:
1223:
1195:
1180:
1160:
1159:
1143:
1137:
1136:
1117:
1100:
1079:
1078:
1012:
995:
979:
978:
944:
927:
911:
910:
857:
841:
840:
809:
793:
792:
742:
726:
725:
677:
661:
660:
642:
627:
575:
568:
528:
527:
508:
507:
477:
476:
457:
456:
437:
436:
417:
416:
383:
379:
374:
373:
332:
331:
264:
263:
257:
223:
215:
196:
169:in his work on
89:
85:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8537:
8535:
8527:
8526:
8521:
8511:
8510:
8507:
8506:
8504:Science Museum
8490:
8478:
8477:External links
8475:
8472:
8471:
8465:Solid geometry
8455:
8440:
8425:
8407:
8392:
8370:
8346:
8344:, 1918, page 5
8333:
8323:Fowler, R. H.
8316:
8300:
8298:, Vol. 5, 1809
8287:
8280:
8252:
8229:
8211:
8204:
8175:
8162:
8150:
8122:(3): 183–192,
8106:
8087:
8086:
8085:
8084:
8080:
8079:
8060:
8059:
8054:
8051:
8050:
8049:
8047:Big O notation
8042:
8039:
8018:
8015:
8012:
8009:
8006:
8003:
8000:
7997:
7994:
7989:
7985:
7964:
7961:
7956:
7953:
7950:
7946:
7925:
7899:
7895:
7874:
7871:
7868:
7863:
7859:
7855:
7852:
7849:
7846:
7843:
7840:
7837:
7834:
7831:
7826:
7823:
7820:
7816:
7812:
7809:
7806:
7803:
7800:
7797:
7794:
7791:
7786:
7782:
7767:
7766:
7755:
7752:
7745:
7741:
7735:
7731:
7725:
7718:
7714:
7708:
7704:
7698:
7691:
7687:
7681:
7677:
7657:
7656:
7645:
7642:
7635:
7631:
7625:
7621:
7615:
7608:
7604:
7598:
7594:
7588:
7581:
7577:
7571:
7567:
7547:
7546:
7535:
7532:
7525:
7521:
7515:
7511:
7505:
7498:
7494:
7488:
7484:
7467:
7466:
7455:
7452:
7447:
7444:
7439:
7436:
7433:
7419:
7418:
7407:
7404:
7397:
7393:
7387:
7383:
7377:
7370:
7366:
7360:
7356:
7327:
7324:
7303:
7300:
7296:
7292:
7288:
7284:
7281:
7278:
7274:
7270:
7266:
7239:
7230:
7212:
7209:
7206:
7203:
7200:
7197:
7194:
7191:
7186:
7183:
7180:
7176:
7172:
7169:
7166:
7163:
7160:
7157:
7153:
7149:
7145:
7141:
7138:
7135:
7132:
7129:
7126:
7122:
7118:
7114:
7085:
7082:
7079:
7076:
7073:
7070:
7067:
7062:
7059:
7056:
7052:
7031:
7028:
7025:
7022:
7019:
7016:
7013:
7009:
7005:
7001:
6997:
6994:
6991:
6988:
6985:
6982:
6978:
6974:
6970:
6949:
6946:
6943:
6940:
6937:
6933:
6929:
6925:
6904:
6901:
6898:
6895:
6892:
6888:
6884:
6880:
6868:
6867:
6856:
6853:
6850:
6847:
6844:
6841:
6838:
6833:
6830:
6827:
6823:
6819:
6816:
6813:
6810:
6807:
6804:
6801:
6797:
6793:
6789:
6785:
6782:
6779:
6776:
6773:
6770:
6767:
6763:
6759:
6755:
6726:
6701:
6690:
6676:
6659:
6653:
6652:
6639:
6635:
6631:
6628:
6625:
6622:
6619:
6616:
6611:
6607:
6603:
6600:
6597:
6594:
6591:
6588:
6585:
6582:
6577:
6574:
6571:
6567:
6563:
6560:
6557:
6554:
6551:
6548:
6543:
6539:
6535:
6532:
6529:
6526:
6523:
6520:
6517:
6421:
6418:
6405:
6402:
6375:
6374:
6361:
6357:
6354:
6351:
6348:
6345:
6340:
6336:
6332:
6329:
6324:
6320:
6313:
6310:
6176:
6173:
6159:
6099:
6098:
6087:
6084:
6081:
6078:
6075:
6072:
6069:
6066:
6063:
6060:
6057:
6008:
6007:
5991:
5987:
5983:
5978:
5974:
5967:
5963:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5936:
5933:
5930:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5905:
5883:
5882:
5866:
5862:
5858:
5853:
5849:
5842:
5838:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5814:
5811:
5808:
5805:
5802:
5798:
5747:
5744:
5741:
5738:
5735:
5732:
5729:
5726:
5723:
5610:
5609:
5598:
5595:
5592:
5589:
5586:
5583:
5580:
5575:
5571:
5567:
5564:
5561:
5558:
5553:
5549:
5545:
5540:
5537:
5534:
5530:
5460:
5457:
5437:
5436:
5385:
5384:
5342:
5282:
5279:
5248:
5247:
5233:
5230:
5227:
5223:
5218:
5215:
5212:
5206:
5203:
5200:
5195:
5192:
5189:
5186:
5181:
5177:
5170:
5167:
5164:
5161:
5158:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5095:
5092:
5072:
5069:
5066:
5046:
5043:
5040:
5037:
5034:
5030:
5026:
5023:
5020:
5017:
5014:
5009:
5005:
5001:
4998:
4995:
4992:
4989:
4986:
4970:
4967:
4966:
4965:
4951:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4927:
4924:
4921:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4895:
4892:
4889:
4883:
4877:
4874:
4871:
4866:
4862:
4858:
4855:
4850:
4846:
4840:
4837:
4834:
4831:
4828:
4823:
4819:
4812:
4809:
4806:
4803:
4800:
4771:
4770:
4763:
4749:
4746:
4741:
4737:
4733:
4726:
4722:
4718:
4712:
4709:
4706:
4703:
4700:
4690:
4687:
4683:
4682:
4671:
4668:
4665:
4662:
4659:
4656:
4646:
4633:
4630:
4625:
4622:
4619:
4616:
4613:
4610:
4605:
4601:
4598:
4595:
4592:
4589:
4584:
4580:
4576:
4570:
4567:
4564:
4561:
4558:
4548:
4538:
4534:
4533:
4520:
4517:
4512:
4509:
4499:
4485:
4482:
4479:
4476:
4471:
4467:
4463:
4458:
4455:
4450:
4446:
4442:
4436:
4433:
4430:
4427:
4424:
4414:
4408:
4404:
4403:
4392:
4389:
4386:
4376:
4362:
4359:
4354:
4350:
4345:
4340:
4337:
4334:
4331:
4328:
4318:
4307:
4304:
4301:
4291:
4287:
4286:
4283:
4280:
4277:
4252:
4249:
4231:
4230:
4218:
4215:
4212:
4207:
4204:
4201:
4198:
4194:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4158:
4155:
4152:
4149:
4145:
4141:
4138:
4127:
4126:
4114:
4111:
4106:
4102:
4099:
4096:
4088:
4085:
4082:
4079:
4075:
4071:
4068:
4064:
4060:
4057:
4054:
4051:
4046:
4043:
4040:
4037:
4033:
4029:
4026:
3972:
3971:
3960:
3957:
3953:
3949:
3946:
3943:
3938:
3934:
3931:
3928:
3925:
3922:
3917:
3913:
3909:
3902:
3896:
3893:
3890:
3887:
3883:
3879:
3876:
3873:
3870:
3867:
3864:
3861:
3858:
3855:
3852:
3847:
3844:
3841:
3838:
3834:
3830:
3827:
3817:
3805:
3802:
3795:
3791:
3786:
3783:
3780:
3777:
3774:
3769:
3765:
3761:
3753:
3750:
3747:
3744:
3740:
3736:
3733:
3729:
3725:
3722:
3719:
3716:
3711:
3708:
3705:
3702:
3698:
3694:
3691:
3614:
3613:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3573:
3570:
3567:
3563:
3549:
3541:
3507:
3504:
3484:
3481:
3466:
3465:
3454:
3450:
3446:
3443:
3440:
3437:
3432:
3429:
3426:
3422:
3408:
3400:
3361:
3358:
3353:
3350:
3349:
3348:
3337:
3334:
3329:
3326:
3319:
3316:
3313:
3310:
3306:
3302:
3292:
3280:
3276:
3273:
3269:
3263:
3260:
3255:
3252:
3248:
3243:
3237:
3234:
3231:
3228:
3224:
3220:
3210:
3198:
3194:
3191:
3188:
3185:
3182:
3179:
3175:
3169:
3166:
3163:
3160:
3156:
3119:) =
3111:An example is
3054:
3053:
3042:
3039:
3035:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2997:
2991:
2988:
2985:
2982:
2978:
2971: or
2966:
2963:
2959:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2921:
2915:
2912:
2909:
2906:
2902:
2813:
2810:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2770:
2767:
2759:error function
2746:
2743:
2738:
2734:
2730:
2727:
2724:
2721:
2718:
2715:
2687:
2686:
2675:
2672:
2666:
2663:
2658:
2654:
2649:
2642:
2639:
2636:
2633:
2629:
2625:
2619:
2616:
2611:
2607:
2602:
2595:
2592:
2589:
2586:
2582:
2561:tends both to
2498:
2497:
2486:
2481:
2478:
2473:
2470:
2467:
2464:
2461:
2458:
2453:
2450:
2447:
2444:
2440:
2417:
2414:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2386:
2383:
2380:
2377:
2373:
2303:
2302:
2290:
2287:
2284:
2281:
2278:
2275:
2270:
2267:
2264:
2261:
2257:
2236:
2233:
2230:
2227:
2224:
2221:
2216:
2213:
2210:
2207:
2203:
2145:
2142:
2129:
2126:
2123:
2103:
2100:
2097:
2093:
2090:
2069:
2066:
2046:
2043:
2023:
2002:
1999:
1974:
1954:
1949:
1945:
1941:
1937:
1934:
1913:
1910:
1907:
1896:
1895:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1838:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1807:
1803:
1799:
1796:
1793:
1787:
1782:
1778:
1763:
1762:
1746:
1742:
1737:
1734:
1731:
1728:
1722:
1719:
1713:
1710:
1707:
1704:
1701:
1695:
1692:
1689:
1686:
1682:
1679:
1655:
1644:
1643:
1631:
1628:
1625:
1621:
1616:
1610:
1607:
1601:
1597:
1594:
1591:
1585:
1582:
1575:
1567:
1563:
1559:
1556:
1552:
1548:
1545:
1542:
1539:
1536:
1529:
1525:
1521:
1518:
1514:
1499:
1498:
1487:
1484:
1481:
1478:
1474:
1469:
1463:
1460:
1454:
1450:
1447:
1444:
1438:
1435:
1428:
1420:
1416:
1412:
1409:
1405:
1401:
1398:
1395:
1392:
1389:
1382:
1378:
1374:
1371:
1367:
1343:
1340:
1337:
1334:
1323:
1322:
1310:
1307:
1304:
1282:
1276:
1273:
1267:
1264:
1261:
1258:
1252:
1249:
1243:
1240:
1237:
1234:
1231:
1201:, even though
1177:
1176:
1163:
1158:
1155:
1152:
1144:
1142:
1139:
1138:
1135:
1132:
1129:
1126:
1118:
1114:
1111:
1106:
1105:
1103:
1098:
1095:
1092:
1089:
1086:
1045:
1044:
1033:
1030:
1027:
1021:
1018:
1015:
1011:
1002:
998:
994:
991:
987:
976:
965:
962:
959:
953:
950:
947:
943:
934:
930:
926:
923:
919:
864:
860:
856:
853:
849:
816:
812:
808:
805:
801:
789:
788:
777:
774:
771:
768:
765:
762:
759:
756:
749:
745:
741:
738:
734:
723:
712:
709:
706:
703:
700:
697:
694:
691:
684:
680:
676:
673:
669:
626:
623:
567:
564:
535:
515:
489:
486:
464:
444:
424:
403:
397:
394:
389:
386:
382:
359:
356:
351:
348:
345:
342:
339:
292:
289:
283:
280:
277:
274:
271:
256:
253:
171:conic sections
60: = 2
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8536:
8525:
8522:
8520:
8517:
8516:
8514:
8505:
8501:
8497:
8494:
8491:
8488:
8484:
8481:
8480:
8476:
8468:
8466:
8459:
8456:
8453:
8452:(1922) p. 271
8451:
8444:
8441:
8436:
8432:
8428:
8422:
8418:
8411:
8408:
8403:
8399:
8395:
8393:0-486-49576-0
8389:
8385:
8381:
8374:
8371:
8367:
8366:0-521-64140-3
8363:
8360:
8356:
8350:
8347:
8343:
8337:
8334:
8330:
8326:
8320:
8317:
8314:
8310:
8304:
8301:
8297:
8291:
8288:
8283:
8281:0-486-44277-2
8277:
8273:
8269:
8265:
8264:
8256:
8253:
8248:
8244:
8240:
8233:
8230:
8226:
8225:
8221:
8215:
8212:
8207:
8201:
8197:
8192:
8191:
8185:
8179:
8176:
8172:
8166:
8163:
8159:
8154:
8151:
8147:
8143:
8139:
8135:
8130:
8129:10.1.1.502.72
8125:
8121:
8117:
8110:
8107:
8103:
8099:
8092:
8089:
8082:
8081:
8077:
8073:
8072:
8067:
8062:
8061:
8057:
8056:
8052:
8048:
8045:
8044:
8040:
8038:
8036:
8032:
8016:
8013:
8007:
8004:
8001:
7998:
7995:
7987:
7983:
7962:
7959:
7954:
7951:
7948:
7944:
7923:
7915:
7897:
7893:
7872:
7869:
7866:
7861:
7857:
7853:
7850:
7844:
7841:
7838:
7835:
7832:
7824:
7821:
7818:
7814:
7810:
7804:
7801:
7798:
7795:
7792:
7784:
7780:
7770:
7753:
7750:
7743:
7739:
7733:
7729:
7723:
7716:
7712:
7706:
7702:
7696:
7689:
7685:
7679:
7675:
7665:
7664:
7663:
7662:
7643:
7640:
7633:
7629:
7623:
7619:
7613:
7606:
7602:
7596:
7592:
7586:
7579:
7575:
7569:
7565:
7555:
7554:
7553:
7552:
7533:
7530:
7523:
7519:
7513:
7509:
7503:
7496:
7492:
7486:
7482:
7472:
7471:
7470:
7453:
7450:
7445:
7442:
7437:
7434:
7431:
7424:
7423:
7422:
7405:
7402:
7395:
7391:
7385:
7381:
7375:
7368:
7364:
7358:
7354:
7344:
7343:
7342:
7341:
7332:
7325:
7323:
7321:
7317:
7301:
7298:
7290:
7282:
7279:
7276:
7268:
7253:
7249:
7242:
7238:
7233:
7229:
7224:
7210:
7207:
7204:
7198:
7195:
7192:
7184:
7181:
7178:
7174:
7170:
7164:
7161:
7158:
7151:
7147:
7143:
7139:
7133:
7130:
7127:
7120:
7116:
7112:
7103:
7083:
7080:
7074:
7071:
7068:
7060:
7057:
7054:
7050:
7029:
7026:
7020:
7017:
7014:
7007:
7003:
6999:
6995:
6989:
6986:
6983:
6976:
6972:
6968:
6944:
6941:
6938:
6931:
6927:
6923:
6899:
6896:
6893:
6886:
6882:
6878:
6854:
6851:
6845:
6842:
6839:
6831:
6828:
6825:
6821:
6817:
6814:
6808:
6805:
6802:
6795:
6791:
6787:
6783:
6780:
6774:
6771:
6768:
6761:
6757:
6753:
6745:
6744:
6743:
6739:
6735:
6729:
6725:
6721:
6717:
6713:
6709:
6704:
6700:
6693:
6689:
6685:
6679:
6675:
6671:
6667:
6662:
6658:
6637:
6633:
6629:
6623:
6620:
6617:
6609:
6605:
6601:
6598:
6595:
6589:
6586:
6583:
6575:
6572:
6569:
6565:
6561:
6555:
6552:
6549:
6541:
6537:
6533:
6527:
6524:
6521:
6515:
6508:
6507:
6506:
6505:
6501:
6497:
6493:
6489:
6484:
6482:
6478:
6474:
6470:
6465:
6463:
6459:
6455:
6451:
6447:
6443:
6435:
6431:
6426:
6419:
6417:
6415:
6411:
6403:
6401:
6399:
6395:
6389:
6385:
6381:
6359:
6355:
6352:
6349:
6346:
6343:
6338:
6334:
6330:
6327:
6322:
6318:
6311:
6308:
6301:
6300:
6299:
6296:
6294:
6290:
6287:. Sometimes
6286:
6283: →
6282:
6278:
6274:
6270:
6266:
6262:
6258:
6254:
6250:
6246:
6242:
6238:
6234:
6230:
6225:
6221:
6217:
6213:
6205:
6201:
6197:
6193:
6189:
6185:
6181:
6174:
6172:
6157:
6149:
6144:
6140:
6136:
6131:
6127:
6124: =
6123:
6118:
6116:
6112:
6108:
6104:
6085:
6076:
6070:
6067:
6064:
6055:
6048:
6047:
6046:
6044:
6040:
6036:
6032:
6028:
6025: =
6024:
6020:
6019:real function
6016:
6011:
5989:
5985:
5981:
5976:
5972:
5961:
5958:
5949:
5943:
5937:
5934:
5931:
5922:
5916:
5910:
5907:
5892:
5891:
5890:
5888:
5864:
5860:
5856:
5851:
5847:
5836:
5833:
5827:
5821:
5818:
5815:
5809:
5803:
5800:
5785:
5784:
5783:
5781:
5777:
5773:
5769:
5765:
5761:
5745:
5742:
5739:
5736:
5733:
5730:
5727:
5724:
5721:
5712:
5710:
5706:
5702:
5698:
5694:
5690:
5686:
5682:
5678:
5674:
5670:
5666:
5662:
5658:
5654:
5650:
5647: =
5646:
5642:
5638:
5633:
5631:
5628: →
5627:
5623:
5619:
5615:
5596:
5590:
5581:
5573:
5569:
5565:
5559:
5551:
5547:
5538:
5532:
5520:
5519:
5518:
5516:
5512:
5508:
5504:
5500:
5496:
5492:
5487:
5483:
5479:
5475:
5465:
5458:
5456:
5454:
5450:
5446:
5442:
5439:For example,
5434:
5430:
5426:
5422:
5418:
5414:
5410:
5406:
5402:
5398:
5394:
5393:
5392:
5390:
5383:
5379:
5375:
5371:
5367:
5363:
5359:
5355:
5351:
5347:
5343:
5340:
5336:
5332:
5328:
5324:
5320:
5316:
5312:
5308:
5304:
5300:
5299:
5298:
5296:
5292:
5288:
5280:
5278:
5276:
5271:
5269:
5265:
5261:
5257:
5253:
5231:
5228:
5225:
5221:
5216:
5213:
5210:
5204:
5201:
5198:
5193:
5190:
5187:
5184:
5179:
5175:
5168:
5162:
5156:
5149:
5148:
5147:
5145:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5070:
5067:
5064:
5041:
5038:
5035:
5028:
5021:
5018:
5015:
5012:
5007:
5003:
4996:
4990:
4984:
4975:
4968:
4946:
4943:
4940:
4931:
4928:
4925:
4919:
4911:
4908:
4905:
4896:
4893:
4890:
4881:
4875:
4872:
4869:
4864:
4860:
4856:
4853:
4848:
4844:
4838:
4835:
4832:
4829:
4826:
4821:
4817:
4810:
4804:
4798:
4791:
4790:
4789:
4787:
4783:
4779:
4768:
4764:
4747:
4744:
4739:
4735:
4731:
4724:
4720:
4716:
4710:
4704:
4698:
4691:
4688:
4685:
4684:
4669:
4666:
4663:
4660:
4657:
4654:
4647:
4631:
4628:
4623:
4620:
4617:
4614:
4611:
4608:
4603:
4599:
4596:
4593:
4590:
4587:
4582:
4578:
4574:
4568:
4562:
4556:
4549:
4546:
4542:
4539:
4536:
4535:
4518:
4515:
4510:
4507:
4500:
4483:
4480:
4477:
4474:
4469:
4465:
4461:
4456:
4453:
4448:
4444:
4440:
4434:
4428:
4422:
4415:
4412:
4409:
4406:
4405:
4390:
4387:
4384:
4377:
4360:
4357:
4352:
4348:
4343:
4338:
4332:
4326:
4319:
4305:
4302:
4299:
4292:
4289:
4288:
4284:
4281:
4278:
4275:
4274:
4268:
4265:
4260:
4258:
4250:
4248:
4247:tends to +∞.
4246:
4241:
4237:
4216:
4213:
4210:
4202:
4196:
4188:
4182:
4179:
4176:
4170:
4164:
4153:
4147:
4139:
4136:
4129:
4128:
4112:
4109:
4104:
4100:
4097:
4094:
4083:
4077:
4069:
4066:
4062:
4055:
4049:
4041:
4035:
4027:
4024:
4017:
4016:
4015:
4012:
4009:) = ln
4008:
4004:
4000:The function
3998:
3997:tends to +∞.
3996:
3992:
3988:
3982:
3978:
3958:
3955:
3951:
3947:
3944:
3941:
3936:
3932:
3929:
3926:
3923:
3920:
3915:
3911:
3907:
3900:
3891:
3885:
3877:
3871:
3868:
3865:
3859:
3853:
3842:
3836:
3828:
3825:
3818:
3803:
3800:
3793:
3789:
3784:
3781:
3778:
3775:
3772:
3767:
3763:
3759:
3748:
3742:
3734:
3731:
3727:
3720:
3714:
3706:
3700:
3692:
3689:
3682:
3681:
3680:
3677:
3673:
3669:
3665:
3661:
3655:
3653:
3649:
3645:
3641:
3636:
3632:
3628:
3623:
3619:
3597:
3594:
3591:
3585:
3579:
3571:
3565:
3547:
3539:
3532:
3531:
3530:
3528:
3524:
3519:
3502:
3479:
3471:
3452:
3448:
3441:
3435:
3430:
3424:
3406:
3398:
3391:
3390:
3389:
3387:
3383:
3379:
3375:
3371:
3367:
3359:
3357:
3351:
3335:
3332:
3327:
3324:
3314:
3308:
3300:
3293:
3278:
3274:
3271:
3267:
3261:
3258:
3253:
3250:
3246:
3241:
3232:
3226:
3218:
3211:
3196:
3192:
3189:
3183:
3177:
3173:
3164:
3158:
3146:
3145:
3144:
3142:
3138:
3134:
3131: =
3130:
3126:
3122:
3118:
3114:
3109:
3107:
3103:
3099:
3094:
3090:
3086:
3081:
3077:
3073:
3068:
3064:
3060:
3040:
3037:
3033:
3026:
3023:
3020:
3017:
3011:
3005:
2999:
2995:
2986:
2980:
2964:
2961:
2957:
2950:
2947:
2944:
2941:
2935:
2929:
2923:
2919:
2910:
2904:
2892:
2891:
2890:
2888:
2883:
2879:
2875:
2870:
2866:
2863:. A function
2862:
2858:
2854:
2850:
2842:
2838:
2834:
2830:
2811:
2808:
2802:
2799:
2796:
2790:
2784:
2775:
2768:
2766:
2764:
2760:
2744:
2736:
2732:
2728:
2722:
2719:
2713:
2706:
2702:
2699:(that has an
2697:
2693:
2673:
2670:
2664:
2661:
2656:
2652:
2647:
2637:
2631:
2623:
2617:
2614:
2609:
2605:
2600:
2590:
2584:
2572:
2571:
2570:
2560:
2556:
2550:
2546:
2539:
2533:
2523:
2514:
2504:
2484:
2479:
2476:
2471:
2465:
2459:
2456:
2448:
2442:
2415:
2412:
2407:
2404:
2398:
2392:
2389:
2381:
2375:
2363:
2362:
2361:
2359:
2354:
2348:
2344:
2341: =
2340:
2336:
2332:
2324:
2320:
2317: =
2316:
2312:
2308:
2288:
2285:
2279:
2273:
2265:
2259:
2234:
2231:
2225:
2219:
2211:
2205:
2193:
2192:
2191:
2189:
2185:
2182: =
2181:
2177:
2174: =
2173:
2167:
2162:
2155:
2150:
2143:
2141:
2127:
2124:
2121:
2098:
2091:
2088:
2064:
2041:
2021:
2000:
1997:
1988:
1972:
1947:
1943:
1935:
1932:
1911:
1908:
1905:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1836:
1830:
1824:
1821:
1818:
1815:
1805:
1797:
1794:
1785:
1780:
1776:
1768:
1767:
1766:
1744:
1740:
1732:
1729:
1720:
1717:
1708:
1705:
1699:
1693:
1687:
1680:
1677:
1669:
1668:
1667:
1653:
1626:
1623:
1619:
1614:
1608:
1605:
1599:
1595:
1592:
1589:
1583:
1580:
1573:
1565:
1561:
1554:
1546:
1540:
1534:
1527:
1523:
1516:
1504:
1503:
1502:
1485:
1479:
1476:
1472:
1467:
1461:
1458:
1452:
1448:
1445:
1442:
1436:
1433:
1426:
1418:
1414:
1407:
1399:
1393:
1387:
1380:
1376:
1369:
1357:
1356:
1355:
1341:
1338:
1335:
1332:
1308:
1305:
1302:
1274:
1271:
1262:
1259:
1256:
1250:
1247:
1241:
1235:
1229:
1222:
1221:
1220:
1217:
1214:
1212:
1208:
1204:
1198:
1193:
1189:
1183:
1156:
1153:
1150:
1140:
1133:
1130:
1127:
1124:
1112:
1109:
1101:
1096:
1090:
1084:
1077:
1076:
1075:
1073:
1069:
1065:
1061:
1057:
1054:The function
1052:
1050:
1028:
1025:
1019:
1016:
1013:
1009:
1000:
996:
989:
977:
960:
957:
951:
948:
945:
941:
932:
928:
921:
909:
908:
907:
905:
901:
897:
893:
888:
886:
882:
862:
858:
851:
838:
834:
814:
810:
803:
775:
769:
766:
760:
754:
747:
743:
736:
724:
710:
704:
701:
695:
689:
682:
678:
671:
659:
658:
657:
653:
649:
645:
640:
636:
632:
624:
622:
620:
616:
612:
608:
604:
600:
596:
592:
586:
582:
578:
573:
565:
563:
561:
557:
553:
549:
533:
513:
505:
487:
484:
462:
442:
422:
401:
395:
392:
387:
384:
380:
357:
354:
349:
343:
337:
328:
326:
317:
313:
309:
290:
287:
281:
275:
269:
261:
254:
252:
250:
246:
241:
239:
235:
231:
226:
221:
213:
207:
203:
199:
195:
191:
187:
183:
179:
174:
172:
168:
164:
160:
156:
151:
149:
145:
141:
137:
133:
129:
125:
119:
83:
79:
70:
63:
59:
55:
51:
46:
40:
33:
19:
8464:
8458:
8449:
8443:
8416:
8410:
8382:, New York:
8379:
8373:
8354:
8349:
8341:
8336:
8324:
8319:
8308:
8303:
8295:
8290:
8262:
8255:
8238:
8232:
8222:
8214:
8189:
8178:
8170:
8169:D.E. Smith,
8165:
8157:
8153:
8119:
8115:
8109:
8101:
8098:"Asymptotes"
8091:
8069:
8034:
7771:
7768:
7660:
7658:
7548:
7468:
7420:
7337:
7319:
7315:
7251:
7247:
7240:
7236:
7231:
7227:
7225:
7097:
6869:
6737:
6733:
6727:
6723:
6719:
6715:
6711:
6707:
6702:
6698:
6691:
6687:
6683:
6677:
6673:
6669:
6660:
6656:
6654:
6503:
6499:
6495:
6491:
6487:
6485:
6472:
6468:
6466:
6446:affine plane
6439:
6407:
6393:
6387:
6383:
6379:
6376:
6297:
6292:
6288:
6284:
6280:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6248:
6244:
6240:
6236:
6232:
6228:
6223:
6219:
6215:
6211:
6208:
6203:
6199:
6195:
6191:
6187:
6183:
6147:
6142:
6138:
6134:
6129:
6125:
6121:
6119:
6114:
6110:
6106:
6102:
6100:
6042:
6038:
6034:
6030:
6026:
6022:
6012:
6009:
5886:
5884:
5779:
5775:
5771:
5767:
5763:
5759:
5713:
5708:
5704:
5700:
5696:
5692:
5688:
5684:
5680:
5676:
5672:
5668:
5664:
5660:
5656:
5652:
5648:
5644:
5640:
5636:
5634:
5629:
5625:
5621:
5617:
5613:
5611:
5514:
5510:
5506:
5502:
5498:
5494:
5485:
5481:
5477:
5473:
5470:
5452:
5448:
5444:
5440:
5438:
5432:
5428:
5424:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5386:
5381:
5377:
5373:
5369:
5365:
5361:
5357:
5353:
5349:
5345:
5338:
5334:
5330:
5326:
5322:
5318:
5314:
5310:
5306:
5302:
5294:
5290:
5286:
5284:
5274:
5272:
5267:
5263:
5259:
5255:
5251:
5249:
5141:
4785:
4781:
4777:
4774:
4540:
4410:
4261:
4254:
4244:
4239:
4235:
4232:
4010:
4006:
4002:
3999:
3994:
3990:
3986:
3980:
3976:
3973:
3675:
3671:
3667:
3663:
3659:
3656:
3651:
3647:
3643:
3639:
3634:
3630:
3626:
3621:
3617:
3615:
3526:
3522:
3520:
3469:
3467:
3385:
3381:
3377:
3373:
3369:
3365:
3363:
3355:
3140:
3136:
3132:
3128:
3124:
3120:
3116:
3112:
3110:
3105:
3101:
3097:
3092:
3088:
3084:
3079:
3075:
3071:
3066:
3062:
3058:
3055:
2886:
2881:
2877:
2873:
2868:
2864:
2860:
2856:
2852:
2848:
2846:
2840:
2836:
2832:
2828:
2695:
2691:
2688:
2558:
2554:
2548:
2544:
2540:
2531:
2521:
2512:
2502:
2500:So the line
2499:
2355:
2346:
2342:
2338:
2334:
2330:
2322:
2318:
2314:
2310:
2306:
2304:
2187:
2183:
2179:
2175:
2171:
2165:
2160:
2159:
1897:
1764:
1645:
1500:
1324:
1218:
1215:
1202:
1196:
1191:
1187:
1181:
1178:
1071:
1067:
1063:
1059:
1055:
1053:
1048:
1046:
903:
899:
895:
891:
889:
884:
880:
836:
832:
790:
651:
647:
643:
638:
634:
630:
628:
618:
614:
610:
606:
602:
598:
594:
584:
580:
576:
569:
555:
551:
547:
503:
329:
321:
315:
311:
255:Introduction
245:in the large
244:
242:
237:
233:
229:
227:
219:
211:
205:
201:
197:
185:
181:
177:
175:
158:
157:ἀσύμπτωτος (
152:
134:coordinates
131:
127:
81:
75:
61:
57:
53:
49:
39:Asymptomatic
8220:"Asymptote"
8066:"Asymptote"
7551:hyperboloid
6666:homogeneous
6430:cubic curve
6117:can be +∞.
5675:-axis is 1/
5254:increases,
4238:= ln
883:approaches
306:graphed on
8513:Categories
8487:PlanetMath
8340:Frost, P.
8307:Frost, P.
8285:, p. 89ff.
8053:References
8029:defines a
7916:of degree
7885:where the
6668:of degree
6452:through a
6222:) →
5491:parametric
5484:) →
3472:is either
2761:, and the
2358:arctangent
2154:arctangent
1211:continuous
595:horizontal
178:horizontal
159:asumptōtos
8502:from the
8483:Asymptote
8463:P. Frost
8438:, p. 121.
8124:CiteSeerX
8076:EMS Press
7952:−
7854:⋯
7822:−
7724:−
7697:−
7614:−
7587:−
7504:−
7438:±
7376:−
7340:hyperbola
7299:≤
7277:≤
7182:−
7081:≠
7058:−
6829:−
6599:⋯
6573:−
6214: : (
6109:), where
6059:↦
5944:γ
5917:γ
5699:-axis is
5594:∞
5536:→
5476: : (
4944:−
4929:−
4909:−
4894:−
4854:−
4827:−
4780:= 0, and
4214:
4206:∞
4200:→
4177:−
4157:∞
4151:→
4098:
4087:∞
4081:→
4045:∞
4039:→
3942:−
3895:∞
3889:→
3866:−
3846:∞
3840:→
3752:∞
3746:→
3710:∞
3704:→
3650:tends to
3592:−
3569:→
3506:∞
3483:∞
3480:−
3428:→
3318:∞
3315:±
3312:→
3272:−
3236:∞
3233:±
3230:→
3190:−
3168:∞
3165:±
3162:→
3135:(that is
3012:−
2990:∞
2987:−
2984:→
2936:−
2914:∞
2908:→
2729:−
2723:
2717:↦
2701:hyperbola
2641:∞
2635:→
2594:∞
2591:−
2588:→
2534:tends to
2515:tends to
2477:π
2460:
2452:∞
2446:→
2413:π
2408:−
2393:
2385:∞
2382:−
2379:→
2349:tends to
2325:tends to
2269:∞
2263:→
2215:∞
2212:−
2209:→
2068:∞
2065:−
2045:∞
1883:…
1831:π
1795:−
1709:
1700:−
1630:∞
1627:−
1596:
1566:−
1558:→
1528:−
1520:→
1483:∞
1449:
1411:→
1373:→
1354:because
1263:
1184:→ 0
1154:≤
1032:∞
1029:−
1017:−
1001:−
993:→
964:∞
949:−
925:→
855:→
815:−
807:→
773:∞
770:±
740:→
708:∞
705:±
683:−
675:→
629:The line
238:asymptote
222:tends to
214:tends to
82:asymptote
8496:Archived
8209:, §4.18.
8186:(1967),
8041:See also
7152:′
7121:′
7008:′
6977:′
6932:′
6887:′
6796:′
6762:′
6730:−1
6718:−
6398:parabola
6251:)), and
6146:, where
5415:), then
5360:), then
5317:), then
5144:dividing
4282:Example
4125:and then
3974:so that
3816:and then
2563:−∞
2327:−∞
2092:′
2080:. Hence
2001:′
1936:′
1681:′
1147:if
1121:if
599:vertical
572:calculus
194:function
182:vertical
8435:2156630
8402:0120551
8311:(1918)
8247:0114163
8146:2690881
7254:- 1 = 0
6444:in the
5659:(where
5389:scaling
5289:=0 for
4769:exists
4686:> 1
4290:< 0
3993:) when
3521:Having
3104:) when
3078:) when
2831:-axis (
2547:) = 1/(
603:oblique
186:oblique
144:tangent
122:) of a
8467:(1875)
8433:
8423:
8400:
8390:
8364:
8313:online
8278:
8245:
8202:
8144:
8126:
6655:where
4264:degree
4003:ƒ
3666:) = (2
3660:ƒ
3616:where
3468:where
2827:, the
2519:, and
2457:arctan
2390:arctan
2337:) has
2313:) has
791:where
648:ƒ
591:limits
581:ƒ
310:. The
202:ƒ
8250:, §8.
8142:JSTOR
6714:) = (
6696:, if
6481:conic
6462:field
6017:of a
6015:graph
5885:if γ(
5489:be a
4788:= 2.
4689:none
3674:+ 1)/
3646:) as
2851:- or
2190:) if
1051:= 1.
637:is a
560:limit
192:of a
190:graph
163:priv.
155:Greek
138:. In
124:curve
80:, an
8421:ISBN
8388:ISBN
8362:ISBN
8276:ISBN
8200:ISBN
8031:cone
7936:and
7912:are
7338:The
7322:=0.
7042:and
6915:and
6209:Let
6202:+4)/
6150:and
5471:Let
5293:(x)=
4537:= 1
4407:= 0
4262:The
4014:has
3679:has
3123:+ 1/
2757:the
2694:↦ 1/
2565:and
2430:and
2152:The
2057:nor
1850:for
1501:and
1128:>
894:) =
601:and
554:and
325:Line
314:and
184:and
8485:at
8268:hdl
8134:doi
6664:is
6390:+ 3
6386:+ 2
5529:lim
5421:cax
5395:If
5344:If
5301:If
4233:So
4193:lim
4144:lim
4074:lim
4032:lim
3983:+ 3
3979:= 2
3882:lim
3833:lim
3739:lim
3697:lim
3670:+ 3
3562:lim
3553:def
3495:or
3421:lim
3412:def
3305:lim
3223:lim
3155:lim
2977:lim
2901:lim
2859:or
2720:exp
2628:lim
2581:lim
2551:+1)
2505:= –
2439:lim
2372:lim
2256:lim
2247:or
2202:lim
2014:at
1989:of
1706:cos
1593:sin
1551:lim
1513:lim
1446:sin
1404:lim
1366:lim
1294:at
1260:sin
1199:= 0
986:lim
918:lim
848:lim
800:lim
733:lim
668:lim
130:or
76:In
8515::
8431:MR
8429:,
8398:MR
8396:,
8386:,
8274:,
8243:MR
8198:,
8140:,
8132:,
8120:72
8118:,
8100:,
8074:,
8068:,
7754:0.
7534:0.
7250:+
6736:,
6722:)
6720:by
6716:ax
6710:,
6686:=
6464:.
6432:,
6428:A
6382:=
6243:),
6198:+3
6194:+2
6186:+2
6141:+
6139:mx
6137:=
5774:),
5651:,
5509:),
5447:)=
5429:cf
5425:cb
5401:ax
5380:)+
5262:=
4484:12
4255:A
4211:ln
4095:ln
3654:.
3633:+
3631:mx
3629:=
3380:+
3378:mx
3376:=
3336:0.
3091:+
3089:mx
3087:=
3065:+
3063:mx
3061:=
3041:0.
2880:+
2878:mx
2876:=
2839:=
2765:.
2674:0.
2567:+∞
2543:ƒ(
2538:.
2536:+∞
2528:/2
2524:=
2517:–∞
2509:/2
2353:.
2351:+∞
2140:.
1186:,
1157:0.
1070:=
898:/(
646:=
633:=
597:,
579:=
251:.
200:=
180:,
150:.
112:oʊ
64:).
8489:.
8368:,
8331:)
8270::
8136::
8017:0
8014:=
8011:)
8008:z
8005:,
8002:y
7999:,
7996:x
7993:(
7988:d
7984:P
7963:0
7960:=
7955:1
7949:d
7945:P
7924:i
7898:i
7894:P
7873:,
7870:0
7867:=
7862:0
7858:P
7851:+
7848:)
7845:z
7842:,
7839:y
7836:,
7833:x
7830:(
7825:2
7819:d
7815:P
7811:+
7808:)
7805:z
7802:,
7799:y
7796:,
7793:x
7790:(
7785:d
7781:P
7751:=
7744:2
7740:c
7734:2
7730:z
7717:2
7713:b
7707:2
7703:y
7690:2
7686:a
7680:2
7676:x
7644:1
7641:=
7634:2
7630:c
7624:2
7620:z
7607:2
7603:b
7597:2
7593:y
7580:2
7576:a
7570:2
7566:x
7531:=
7524:2
7520:b
7514:2
7510:y
7497:2
7493:a
7487:2
7483:x
7454:.
7451:x
7446:a
7443:b
7435:=
7432:y
7406:1
7403:=
7396:2
7392:b
7386:2
7382:y
7369:2
7365:a
7359:2
7355:x
7320:x
7316:x
7302:1
7295:|
7291:y
7287:|
7283:,
7280:1
7273:|
7269:x
7265:|
7252:y
7248:x
7241:n
7237:P
7232:n
7228:P
7211:,
7208:0
7205:=
7202:)
7199:a
7196:,
7193:b
7190:(
7185:1
7179:n
7175:P
7171:=
7168:)
7165:a
7162:,
7159:b
7156:(
7148:y
7144:Q
7140:=
7137:)
7134:a
7131:,
7128:b
7125:(
7117:x
7113:Q
7084:0
7078:)
7075:a
7072:,
7069:b
7066:(
7061:1
7055:n
7051:P
7030:0
7027:=
7024:)
7021:a
7018:,
7015:b
7012:(
7004:y
7000:Q
6996:=
6993:)
6990:a
6987:,
6984:b
6981:(
6973:x
6969:Q
6948:)
6945:a
6942:,
6939:b
6936:(
6928:y
6924:Q
6903:)
6900:a
6897:,
6894:b
6891:(
6883:x
6879:Q
6855:0
6852:=
6849:)
6846:a
6843:,
6840:b
6837:(
6832:1
6826:n
6822:P
6818:+
6815:y
6812:)
6809:a
6806:,
6803:b
6800:(
6792:y
6788:Q
6784:+
6781:x
6778:)
6775:a
6772:,
6769:b
6766:(
6758:x
6754:Q
6740:)
6738:y
6734:x
6732:(
6728:n
6724:Q
6712:y
6708:x
6706:(
6703:n
6699:P
6692:n
6688:P
6684:Q
6678:n
6674:P
6670:k
6661:k
6657:P
6638:0
6634:P
6630:+
6627:)
6624:y
6621:,
6618:x
6615:(
6610:1
6606:P
6602:+
6596:+
6593:)
6590:y
6587:,
6584:x
6581:(
6576:1
6570:n
6566:P
6562:+
6559:)
6556:y
6553:,
6550:x
6547:(
6542:n
6538:P
6534:=
6531:)
6528:y
6525:,
6522:x
6519:(
6516:P
6504:n
6500:P
6496:y
6494:,
6492:x
6490:(
6488:P
6473:n
6469:n
6388:x
6384:x
6380:y
6360:x
6356:4
6353:+
6350:x
6347:3
6344:+
6339:2
6335:x
6331:2
6328:+
6323:3
6319:x
6312:=
6309:y
6293:A
6289:B
6285:b
6281:t
6277:B
6273:t
6271:(
6269:A
6265:A
6261:B
6257:A
6253:B
6249:t
6247:(
6245:y
6241:t
6239:(
6237:x
6233:t
6231:(
6229:A
6224:R
6220:b
6218:,
6216:a
6212:A
6204:x
6200:x
6196:x
6192:x
6188:x
6184:x
6158:n
6148:m
6143:n
6135:y
6130:c
6126:c
6122:x
6115:b
6111:a
6107:b
6105:,
6103:a
6086:.
6083:)
6080:)
6077:t
6074:(
6071:f
6068:,
6065:t
6062:(
6056:t
6043:x
6041:(
6039:ƒ
6037:,
6035:x
6031:x
6029:(
6027:ƒ
6023:y
5990:2
5986:b
5982:+
5977:2
5973:a
5966:|
5962:c
5959:+
5956:)
5953:)
5950:t
5947:(
5941:(
5938:y
5935:b
5932:+
5929:)
5926:)
5923:t
5920:(
5914:(
5911:x
5908:a
5904:|
5887:t
5865:2
5861:b
5857:+
5852:2
5848:a
5841:|
5837:c
5834:+
5831:)
5828:t
5825:(
5822:y
5819:b
5816:+
5813:)
5810:t
5807:(
5804:x
5801:a
5797:|
5780:t
5778:(
5776:y
5772:t
5770:(
5768:x
5764:t
5762:(
5760:A
5746:0
5743:=
5740:c
5737:+
5734:y
5731:b
5728:+
5725:x
5722:a
5709:y
5705:t
5701:t
5697:y
5693:t
5689:y
5685:x
5681:t
5677:t
5673:x
5669:t
5665:x
5661:t
5657:t
5653:y
5649:t
5645:x
5641:x
5637:y
5630:b
5626:t
5622:t
5620:(
5618:A
5614:A
5597:.
5591:=
5588:)
5585:)
5582:t
5579:(
5574:2
5570:y
5566:+
5563:)
5560:t
5557:(
5552:2
5548:x
5544:(
5539:b
5533:t
5515:t
5513:(
5511:y
5507:t
5505:(
5503:x
5499:t
5497:(
5495:A
5486:R
5482:b
5480:,
5478:a
5474:A
5453:y
5449:e
5445:x
5443:(
5441:f
5435:)
5433:x
5431:(
5423:+
5419:=
5417:y
5413:x
5411:(
5409:f
5405:b
5403:+
5399:=
5397:y
5382:k
5378:x
5376:(
5374:f
5370:k
5368:+
5366:c
5364:=
5362:y
5358:x
5356:(
5354:f
5350:c
5348:=
5346:y
5341:)
5339:h
5337:-
5335:x
5333:(
5331:f
5327:h
5325:+
5323:a
5321:=
5319:x
5315:x
5313:(
5311:f
5307:a
5305:=
5303:x
5295:e
5291:f
5287:y
5275:x
5268:x
5264:x
5260:y
5256:f
5252:x
5232:1
5229:+
5226:x
5222:1
5217:+
5214:x
5211:=
5205:1
5202:+
5199:x
5194:1
5191:+
5188:x
5185:+
5180:2
5176:x
5169:=
5166:)
5163:x
5160:(
5157:f
5127:6
5124:,
5121:5
5118:,
5115:4
5112:,
5109:3
5106:,
5103:2
5100:,
5097:1
5094:=
5091:x
5071:x
5068:=
5065:y
5045:)
5042:1
5039:+
5036:x
5033:(
5029:/
5025:)
5022:1
5019:+
5016:x
5013:+
5008:2
5004:x
5000:(
4997:=
4994:)
4991:x
4988:(
4985:f
4950:)
4947:2
4941:x
4938:(
4935:)
4932:1
4926:x
4923:(
4920:x
4915:)
4912:3
4906:x
4903:(
4900:)
4897:2
4891:x
4888:(
4882:=
4876:x
4873:2
4870:+
4865:2
4861:x
4857:3
4849:3
4845:x
4839:6
4836:+
4833:x
4830:5
4822:2
4818:x
4811:=
4808:)
4805:x
4802:(
4799:f
4786:x
4782:x
4778:x
4748:1
4745:+
4740:2
4736:x
4732:3
4725:4
4721:x
4717:2
4711:=
4708:)
4705:x
4702:(
4699:f
4670:3
4667:+
4664:x
4661:2
4658:=
4655:y
4632:x
4629:5
4624:+
4621:3
4618:+
4615:x
4612:2
4609:=
4604:x
4600:5
4597:+
4594:x
4591:3
4588:+
4583:2
4579:x
4575:2
4569:=
4566:)
4563:x
4560:(
4557:f
4541:y
4519:3
4516:2
4511:=
4508:y
4481:+
4478:x
4475:+
4470:2
4466:x
4462:3
4457:7
4454:+
4449:2
4445:x
4441:2
4435:=
4432:)
4429:x
4426:(
4423:f
4411:y
4391:0
4388:=
4385:y
4361:1
4358:+
4353:2
4349:x
4344:1
4339:=
4336:)
4333:x
4330:(
4327:f
4306:0
4303:=
4300:y
4245:x
4240:x
4236:y
4217:x
4203:+
4197:x
4189:=
4186:)
4183:x
4180:m
4174:)
4171:x
4168:(
4165:f
4162:(
4154:+
4148:x
4140:=
4137:n
4113:0
4110:=
4105:x
4101:x
4084:+
4078:x
4070:=
4067:x
4063:/
4059:)
4056:x
4053:(
4050:f
4042:+
4036:x
4028:=
4025:m
4011:x
4007:x
4005:(
3995:x
3991:x
3989:(
3987:ƒ
3981:x
3977:y
3959:3
3956:=
3952:)
3948:x
3945:2
3937:x
3933:1
3930:+
3927:x
3924:3
3921:+
3916:2
3912:x
3908:2
3901:(
3892:+
3886:x
3878:=
3875:)
3872:x
3869:m
3863:)
3860:x
3857:(
3854:f
3851:(
3843:+
3837:x
3829:=
3826:n
3804:2
3801:=
3794:2
3790:x
3785:1
3782:+
3779:x
3776:3
3773:+
3768:2
3764:x
3760:2
3749:+
3743:x
3735:=
3732:x
3728:/
3724:)
3721:x
3718:(
3715:f
3707:+
3701:x
3693:=
3690:m
3676:x
3672:x
3668:x
3664:x
3662:(
3652:a
3648:x
3644:x
3642:(
3640:ƒ
3635:n
3627:y
3622:m
3618:a
3601:)
3598:x
3595:m
3589:)
3586:x
3583:(
3580:f
3577:(
3572:a
3566:x
3548:=
3540:n
3527:n
3523:m
3503:+
3470:a
3453:x
3449:/
3445:)
3442:x
3439:(
3436:f
3431:a
3425:x
3407:=
3399:m
3386:m
3382:n
3374:y
3370:x
3368:(
3366:f
3333:=
3328:x
3325:1
3309:x
3301:=
3279:]
3275:x
3268:)
3262:x
3259:1
3254:+
3251:x
3247:(
3242:[
3227:x
3219:=
3197:]
3193:x
3187:)
3184:x
3181:(
3178:f
3174:[
3159:x
3141:n
3137:m
3133:x
3129:y
3125:x
3121:x
3117:x
3115:(
3113:ƒ
3106:x
3102:x
3100:(
3098:ƒ
3093:n
3085:y
3080:x
3076:x
3074:(
3072:ƒ
3067:n
3059:y
3038:=
3034:]
3030:)
3027:n
3024:+
3021:x
3018:m
3015:(
3009:)
3006:x
3003:(
3000:f
2996:[
2981:x
2965:0
2962:=
2958:]
2954:)
2951:n
2948:+
2945:x
2942:m
2939:(
2933:)
2930:x
2927:(
2924:f
2920:[
2911:+
2905:x
2887:m
2885:(
2882:n
2874:y
2869:x
2867:(
2865:ƒ
2853:y
2849:x
2841:x
2837:y
2833:x
2829:y
2812:x
2809:1
2803:+
2800:x
2797:=
2794:)
2791:x
2788:(
2785:f
2745:,
2742:)
2737:2
2733:x
2726:(
2714:x
2696:x
2692:x
2671:=
2665:1
2662:+
2657:2
2653:x
2648:1
2638:+
2632:x
2624:=
2618:1
2615:+
2610:2
2606:x
2601:1
2585:x
2559:x
2555:y
2549:x
2545:x
2532:x
2526:π
2522:y
2513:x
2507:π
2503:y
2485:.
2480:2
2472:=
2469:)
2466:x
2463:(
2449:+
2443:x
2416:2
2405:=
2402:)
2399:x
2396:(
2376:x
2347:x
2343:c
2339:y
2335:x
2333:(
2331:ƒ
2323:x
2319:c
2315:y
2311:x
2309:(
2307:ƒ
2301:.
2289:c
2286:=
2283:)
2280:x
2277:(
2274:f
2266:+
2260:x
2235:c
2232:=
2229:)
2226:x
2223:(
2220:f
2206:x
2188:x
2186:(
2184:ƒ
2180:y
2176:c
2172:y
2166:x
2128:0
2125:=
2122:x
2102:)
2099:x
2096:(
2089:f
2042:+
2022:0
1998:f
1973:0
1953:)
1948:n
1944:x
1940:(
1933:f
1912:0
1909:=
1906:x
1880:,
1877:2
1874:,
1871:1
1868:,
1865:0
1862:=
1859:n
1837:,
1828:)
1825:1
1822:+
1819:n
1816:2
1813:(
1806:n
1802:)
1798:1
1792:(
1786:=
1781:n
1777:x
1761:.
1745:2
1741:x
1736:)
1733:1
1730:+
1727:)
1721:x
1718:1
1712:(
1703:(
1694:=
1691:)
1688:x
1685:(
1678:f
1654:f
1642:.
1624:=
1620:)
1615:)
1609:x
1606:1
1600:(
1590:+
1584:x
1581:1
1574:(
1562:0
1555:x
1547:=
1544:)
1541:x
1538:(
1535:f
1524:0
1517:x
1486:,
1480:+
1477:=
1473:)
1468:)
1462:x
1459:1
1453:(
1443:+
1437:x
1434:1
1427:(
1419:+
1415:0
1408:x
1400:=
1397:)
1394:x
1391:(
1388:f
1381:+
1377:0
1370:x
1342:,
1339:0
1336:=
1333:x
1321:.
1309:0
1306:=
1303:x
1281:)
1275:x
1272:1
1266:(
1257:+
1251:x
1248:1
1242:=
1239:)
1236:x
1233:(
1230:f
1203:ƒ
1197:x
1192:x
1190:(
1188:ƒ
1182:x
1151:x
1141:5
1134:,
1131:0
1125:x
1113:x
1110:1
1102:{
1097:=
1094:)
1091:x
1088:(
1085:f
1072:a
1068:x
1064:a
1060:x
1058:(
1056:ƒ
1049:x
1026:=
1020:1
1014:x
1010:x
997:1
990:x
961:+
958:=
952:1
946:x
942:x
933:+
929:1
922:x
904:x
900:x
896:x
892:x
885:a
881:x
863:+
859:a
852:x
837:a
833:x
811:a
804:x
776:,
767:=
764:)
761:x
758:(
755:f
748:+
744:a
737:x
711:,
702:=
699:)
696:x
693:(
690:f
679:a
672:x
654:)
652:x
650:(
644:y
635:a
631:x
619:x
615:x
611:x
607:x
587:)
585:x
583:(
577:y
556:y
552:x
548:y
534:y
514:x
504:x
488:x
485:1
463:x
443:y
423:x
402:)
396:x
393:1
388:,
385:x
381:(
358:x
355:1
350:=
347:)
344:x
341:(
338:f
316:y
312:x
291:x
288:1
282:=
279:)
276:x
273:(
270:f
220:x
212:x
208:)
206:x
204:(
198:y
132:y
128:x
118:/
115:t
109:t
106:p
103:m
100:ɪ
97:s
94:æ
91:ˈ
88:/
84:(
62:x
58:y
54:x
50:y
41:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
