# Limit as x &#x2192;<!-- → --> 2 of cos &#x2061;<!-- ⁡ -->

Limit as $x\to 2$ of $\frac{\mathrm{cos}\left(\frac{\pi }{x}\right)}{x-2}$
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Jayvion Mclaughlin
HINT:
$\mathrm{cos}\frac{\pi }{x}=\mathrm{sin}\left(\frac{\pi }{2}-\frac{\pi }{x}\right)=\mathrm{sin}\frac{\pi \left(x-2\right)}{2x}$
Now set $\frac{\pi \left(x-2\right)}{2x}=y$ and use $\underset{h\to a}{lim}\frac{\mathrm{sin}\left(h-a\right)}{h-a}=1$

Jameson Lucero
Let $\frac{1}{x}=y$ then
$\begin{array}{rl}\underset{x\to 2}{lim}\frac{\mathrm{cos}\left(\frac{\pi }{x}\right)}{x-2}& =\underset{y\to \frac{1}{2}}{lim}\frac{\mathrm{sin}\left(\frac{\pi }{2}-\pi y\right)}{\frac{1}{y}-2}\\ & =\underset{y\to \frac{1}{2}}{lim}\frac{\mathrm{sin}\pi \left(\frac{1}{2}-y\right)}{\pi \left(\frac{1}{2}-y\right)}\frac{\left(\frac{1}{2}-y\right)}{1-2y}\pi y\\ & =\frac{\pi }{4}\end{array}$