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limits_extensions.html
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<div class="span10 offset1">
<h1>Limits, issues, extensions of the concept</h1>
<p>The limit of a function at <span class="math">$c$</span> need not exist for one of many different reasons. Some of these reasons can be handled with extensions to the concept of the limit, others are just problematic in terms of limits. This section covers examples of each.</p>
<p>Let's begin with a function that is just problematic. Consider</p>
<p class="math">\[
~
f(x) = \sin(1/x)
~
\]</p>
<p>As this is a composition of nice functions it will have a limit everywhere except possibly when <span class="math">$x=0$</span>, as then <span class="math">$1/x$</span> may not have a limit. So rather than talk about where it is nice, let's consider the question of whether a limit exists at <span class="math">$c=0$</span>.</p>
<p>A graph shows the issue:</p>
<div id="041a6599-f4e3-4495-85c7-5b203fbecede" style="width:576px;height:384px;"></div>
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