limit - Maple Programming Help

limit

calculate limit

Limit

inert form of limit

Calling Sequence

 limit(f, x=a) $\underset{x→a}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f$ Limit(f, x=a) $\underset{x→a}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}f$ limit(f, x=a, dir) Limit(f, x=a, dir)

Parameters

 f - algebraic expression x - name a - algebraic expression; limit point, possibly infinity, or -infinity dir - (optional) symbol; direction chosen from: left, right, real, or complex

Description

 • The limit(f, x=a, dir) function attempts to compute the limiting value of f as x approaches a.
 • You can enter the command limit using either the 1-D or 2-D calling sequence.
 • If dir is not specified, the limit is the real bidirectional limit, except in the case where the limit point is infinity or -infinity, in which case the limit is from the left to infinity and from the right to -infinity. For help with directional limits, see limit/dir.
 • The output from limit can be a range (meaning a bounded result) or an algebraic expression, possibly containing infinity. For further help with the return type, see limit/return.
 • To compute a limit in a multidimensional space, specify a set of points as the second argument. For more information, see limit/multi.
 • Most limits are resolved by computing series. By increasing the value of the global variable Order, the ability of limit to solve problems with significant cancellation improves.
 Note: Since Limit does not try to evaluate or check the existence of the limit of the expression, it can lead to incorrect transformations. Therefore, the use of limit is more reliable. This is demonstrated by the last two examples.
 • If Maple cannot find a closed form for the limit, the function calling sequence is returned.
 • The capitalized function name Limit is the inert limit function, which returns unevaluated.  It appears gray so that it is easily distinguished from a returned limit calling sequence.

Examples

The inert Limit function returns unevaluated.

 > $\underset{x→0}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}$
 $\underset{{x}{→}{0}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{{1}}{{x}}$ (1)
 > $\underset{x→-\mathrm{∞}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\mathrm{ln}\left({x}^{2}\right)$
 $\underset{{x}{→}{-}{\mathrm{∞}}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{ln}}{}\left({{x}}^{{2}}\right)$ (2)
 > $\underset{x→3}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}$
 $\frac{{1}}{{3}}$ (3)
 > $\underset{x→\mathrm{∞}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{x}^{2}$
 ${\mathrm{∞}}$ (4)
 > $\underset{x→0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{\mathrm{sin}\left(x\right)}{x}$
 ${1}$ (5)
 > $\underset{x→\mathrm{∞}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{ⅇ}^{x}$
 ${\mathrm{∞}}$ (6)
 > $\underset{x→-\mathrm{∞}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{ⅇ}^{x}$
 ${0}$ (7)
 > $\underset{x→\mathrm{∞}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{ⅇ}^{{x}^{2}}\left(1-\mathrm{erf}\left(x\right)\right)$
 ${0}$ (8)
 > $\underset{x→\mathrm{∞}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}$
 ${0}$ (9)

To use the real and complex arguments, the limit command must be written in 1-D or 2-D command form and not in mathematical notation. See Entering Commands in 2-D Math for more information.

 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{real}\right)$
 ${\mathrm{undefined}}$ (10)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{complex}\right)$
 ${\mathrm{∞}}{-}{\mathrm{∞}}{}{I}$ (11)

 > $\underset{x→{0}^{+}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}$
 ${\mathrm{∞}}$ (12)
 > $\underset{x→{0}^{-}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}$
 ${-}{\mathrm{∞}}$ (13)
 > $g≔\mathrm{piecewise}\left(x<3,{x}^{2}-6,3\le x,2x-1\right)$
 ${g}{≔}{{}\begin{array}{cc}{{x}}^{{2}}{-}{6}& {x}{<}{3}\\ {2}{}{x}{-}{1}& {3}{\le }{x}\end{array}$ (14)
 > $\underset{x→3}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}g$
 ${\mathrm{undefined}}$ (15)
 > $\underset{x→{3}^{+}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}g$
 ${5}$ (16)
 > $\underset{x→{3}^{-}}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}g$
 ${3}$ (17)

The use of the inert Limit function can lead to errors.

 > $\mathrm{combine}\left(\left(\underset{x→0}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}\right)\left(\underset{x→0}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}x\right)\right)$
 $\underset{{x}{→}{0}}{{lim}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{1}$ (18)
 > $\mathrm{combine}\left(\left(\underset{x→0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\frac{1}{x}\right)\left(\underset{x→0}{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}x\right)\right)$
 ${\mathrm{undefined}}$ (19)

References

 Geddes, K. O., and Gonnet, G. H. "A New Algorithm for Computing Symbolic Limits Using Hierarchical Series." In Proceedings of ISSAC '88, pp. 490-495. Edited by Patrizia M. Gianni. Berlin: Springer-Verlag, 1988.

Compatibility

 • The limit command was updated in Maple 2016; see Advanced Math.