Question

Evaluate the following limits:

${lim}_{x\to \frac{\pi }{2}}(\frac{\pi }{2}-x)tanx$

Answer 1

${lim}_{x\to \frac{\pi }{2}}(\frac{\pi }{2}-x)tanx$

Let $y=\frac{\pi }{2}-x$

as $x\to \frac{\pi }{2},y\to 0$

$={lim}_{y\to 0}ytan(\frac{\pi }{2}-y)$

$={lim}_{y\to 0}ycoty$

$={lim}_{y\to 0}y\frac{cosy}{siny}$

$={lim}_{y\to 0}cosy{lim}_{y\to 0}\frac{y}{siny}$

$={lim}_{y\to 0}cosy\frac{1}{{lim}_{y\to 0}\frac{siny}{y}}$

$=1\times \frac{1}{1}=1$