The limit(f,x=a) function computes the limiting value of f as x approaches a.

The limit function of the MultiSeries package is intended to be used in the same manner as the top-level limit function. However, its output cannot be a range.

The default direction of the limit is two-sided along a line through a and parallel to the real axis. This can be changed by supplying the optional argument dir:

If dir is left or right, a one-sided limit along a ray originating at a and parallel to the real axis is computed.

Specifying real as third argument is equivalent to the default behavior when no direction is specified at all, i.e., a two-sided limit parallel to the real axis.

If dir is complex, a limit is computed that is valid in some open neighborhood of a in the complex plane, typically a disc centered at a.

If dir is fromtheorigin, a limit is computed that is valid along a ray originating at the origin in the complex plane, unless a is 0, in which case the limit is computed from the right. This corresponds to the default behavior of multiseries.

The function limit invokes MultiSeries[multiseries] for its computation. In rare cases, it might be necessary to increase the value of the global variable Order in order to improve the ability of limit to solve problems with significant cancellation. This is made explicit by an error message coming from multiseries.

It can also happen that the result is wrong because Testzero failed to recognize that the leading coefficient of a multiseries expansion happens to be 0. In those cases, it is necessary to modify this environment variable (see Testzero).

$\mathrm{with}\left(\mathrm{MultiSeries}\,\mathrm{limit}\right)\:$

$\mathrm{limit}\left(\frac{\sin \left(x\right)}{x}\,x=0\right)$

$1$

$\mathrm{limit}\left(\exp \left(x\right)\,x=\mathrm{\infty }\right)$

$\mathrm{\infty }$

$\mathrm{limit}\left(\exp \left(x\right)\,x=-\mathrm{\infty }\right)$

$0$

$\mathrm{limit}\left(\frac{1}{x}\,x=0\right)$

$\mathrm{undefined}$

$f≔{\left(\frac{\sin \left(x\right)}{x}\right)}^{\frac{\sin \left(x\right)}{x\left(1-\cos \left(x\right)\right)}}$

$f≔{\left(\frac{\sin \left(x\right)}{x}\right)}^{\frac{\sin \left(x\right)}{x\left(1-\cos \left(x\right)\right)}}$

$\mathrm{limit}\left(f\,x=0\right)$

${\ⅇ}^{-\frac{1}{3}}$

$\mathrm{limit}\left(\sin \left(x\right)\,x=\mathrm{\infty }\right)$

$\mathrm{undefined}$