directional - Maple Help

limit/dir

Directional limits

Calling Sequence

 limit(f, x=a, left) $\underset{x\to a-}{lim}f$ limit(f, x=a, right) $\underset{x\to a+}{lim}f$ limit(f, x=a, dir)

Parameters

 f - algebraic expression x=a - equation, where x is a name and a is the limit point dir - (optional) direction, can be left, right, real , or complex

Description

 • If dir is left or right, the limit is a directional limit, taken from the left or right, respectively. If dir is real, the limit is the bidirectional real limit. If dir is complex, the limit is omni-directional, from all complex directions to the point.
 • 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 or from the right to -infinity respectively.
 • If dir is complex, the limit point infinity denotes complex infinity, that is, all infinities in the complex plane. If dir is real, the limit point infinity denotes both positive and negative infinity, and the limit is done bidirectionally. Otherwise, the limit point infinity denotes positive infinity, and -infinity denotes negative infinity.
 • You can enter the command limit using either the 1-D or 2-D calling sequence.

Examples

 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=\mathrm{\infty }\right)$
 ${\mathrm{\infty }}$ (1)
 > $\mathrm{limit}\left(\mathrm{exp}\left(x\right),x=-\mathrm{\infty }\right)$
 ${0}$ (2)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{right}\right)$
 ${\mathrm{\infty }}$ (3)
 > $\mathrm{limit}\left(\frac{1}{x},x=0,\mathrm{left}\right)$
 ${-}{\mathrm{\infty }}$ (4)
 > $\mathrm{limit}\left(\frac{1}{x},x=0\right)$
 ${\mathrm{undefined}}$ (5)
 > $\mathrm{limit}\left(-x,x=\mathrm{\infty }\right)$
 ${-}{\mathrm{\infty }}$ (6)

Examples using real or complex options

 To use the real and complex arguments, the limit command must be written in 1-D Math notation.
 > limit(1/x, x=0, real);
 ${\mathrm{undefined}}$ (7)
 > limit(1/x, x=0, complex);
 ${\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}$ (8)
 > limit(1/x, x=infinity, real);
 ${0}$ (9)
 > limit(-x, x=infinity, complex);
 ${\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}$ (10)
 > limit(exp(x), x=infinity,real);
 ${\mathrm{undefined}}$ (11)