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<h1 class="chapter"> 18. Limits </h1>
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<tr><td align="left" valign="top"><a href="#SEC72">18.1 Functions and Variables for Limits</a></td><td> </td><td align="left" valign="top">
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<p><a name="Item_003a-Functions-and-Variables-for-Limits"></a>
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<h2 class="section"> 18.1 Functions and Variables for Limits </h2>
<p><a name="Item_003a-lhospitallim"></a>
</p><dl>
<dt><u>Option variable:</u> <b>lhospitallim</b>
<a name="IDX675"></a>
</dt>
<dd><p>Default: 4
</p>
<p><code>lhospitallim</code> is the maximum number of times L'Hospital's
rule is used in <code>limit</code>. This prevents infinite looping in cases like
<code>limit (cot(x)/csc(x), x, 0)</code>.
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<p><a name="Item_003a-limit"></a>
</p><dl>
<dt><u>Function:</u> <b>limit</b><i> (<var>expr</var>, <var>x</var>, <var>val</var>, <var>dir</var>)</i>
<a name="IDX676"></a>
</dt>
<dt><u>Function:</u> <b>limit</b><i> (<var>expr</var>, <var>x</var>, <var>val</var>)</i>
<a name="IDX677"></a>
</dt>
<dt><u>Function:</u> <b>limit</b><i> (<var>expr</var>)</i>
<a name="IDX678"></a>
</dt>
<dd><p>Computes the limit of <var>expr</var> as the real variable
<var>x</var> approaches the value <var>val</var> from the direction <var>dir</var>. <var>dir</var> may have the
value <code>plus</code> for a limit from above, <code>minus</code> for a limit from below, or
may be omitted (implying a two-sided limit is to be computed).
</p>
<p><code>limit</code> uses the
following special symbols: <code>inf</code> (positive infinity) and <code>minf</code> (negative
infinity). On output it may also use <code>und</code> (undefined), <code>ind</code> (indefinite
but bounded) and <code>infinity</code> (complex infinity).
</p>
<p><code>infinity</code> (complex infinity) is returned when the limit of
the absolute value of the expression is positive infinity, but
the limit of the expression itself is not positive infinity or
negative infinity. This includes cases where the limit of the
complex argument is a constant, as in <code>limit(log(x), x, minf)</code>,
cases where the complex argument oscillates, as in
<code>limit((-2)^x, x, inf)</code>, and cases where the complex
argument is different for either side of a two-sided limit, as in
<code>limit(1/x, x, 0)</code> and <code>limit(log(x), x, 0)</code>.
</p>
<p><code>lhospitallim</code> is the maximum number of times L'Hospital's rule
is used in <code>limit</code>. This prevents infinite looping in cases like
<code>limit (cot(x)/csc(x), x, 0)</code>.
</p>
<p><code>tlimswitch</code> when true will allow the <code>limit</code> command to use
Taylor series expansion when necessary.
</p>
<p><code>limsubst</code> prevents <code>limit</code> from attempting substitutions on
unknown forms. This is to avoid bugs like <code>limit (f(n)/f(n+1), n, inf)</code>
giving 1. Setting <code>limsubst</code> to <code>true</code> will allow such
substitutions.
</p>
<p><code>limit</code> with one argument is often called upon to simplify constant expressions,
for example, <code>limit (inf-1)</code>.
</p>
<p><code>example (limit)</code> displays some examples.
</p>
<p>For the method see Wang, P., "Evaluation of Definite Integrals by Symbolic
Manipulation", Ph.D. thesis, MAC TR-92, October 1971.
</p>
<div class=categorybox>
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</dd></dl>
<p><a name="Item_003a-limsubst"></a>
</p><dl>
<dt><u>Option variable:</u> <b>limsubst</b>
<a name="IDX679"></a>
</dt>
<dd><p>default value: <code>false</code> - prevents <code>limit</code> from attempting substitutions on
unknown forms. This is to avoid bugs like <code>limit (f(n)/f(n+1), n, inf)</code>
giving 1. Setting <code>limsubst</code> to <code>true</code> will allow such
substitutions.
</p>
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</dd></dl>
<p><a name="Item_003a-tlimit"></a>
</p><dl>
<dt><u>Function:</u> <b>tlimit</b><i> (<var>expr</var>, <var>x</var>, <var>val</var>, <var>dir</var>)</i>
<a name="IDX680"></a>
</dt>
<dt><u>Function:</u> <b>tlimit</b><i> (<var>expr</var>, <var>x</var>, <var>val</var>)</i>
<a name="IDX681"></a>
</dt>
<dt><u>Function:</u> <b>tlimit</b><i> (<var>expr</var>)</i>
<a name="IDX682"></a>
</dt>
<dd><p>Take the limit of the Taylor series expansion of <code>expr</code> in <code>x</code>
at <code>val</code> from direction <code>dir</code>.
</p>
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<p><a name="Item_003a-tlimswitch"></a>
</p><dl>
<dt><u>Option variable:</u> <b>tlimswitch</b>
<a name="IDX683"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>tlimswitch</code> is <code>true</code>, the <code>limit</code> command will
use a Taylor series expansion if the limit of the input expression cannot be computed directly.
This allows evaluation of limits such as <code>limit(x/(x-1)-1/log(x),x,1,plus)</code>.
When <code>tlimswitch</code> is <code>false</code> and the limit of input expression
cannot be computed directly, <code>limit</code> will return an unevaluated limit expression.
</p>
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