Question

${lim}_{x\to 1}(1-x)\tan \left(\frac{\pi x}{2}\right)$ is equal to -

• A. $\frac{2}{\pi }$
• B. $\frac{\pi }{2}$
• C. $\pi$
• D. none of these

Answer 1

Answer: (A)

Let $t=1-x$
$\Rightarrow x=1-t$
$x\to 1$
$\Rightarrow t=0$
Now ${lim}_{t\to 0}t\tan \left(\frac{\pi }{2}\right(1-t\left)\right)$
$={lim}_{t\to 0}t\tan (\frac{\pi }{2}-\frac{\pi }{2}t)$
$={lim}_{t\to 0}t\cot (\frac{\pi }{2}-t)$
$={lim}_{t\to 0}\frac{t}{\tan \frac{\pi }{2}t}$
Now multiply & divide the above expression by $\frac{\pi }{2}$ we get
$={lim}_{t\to 0}\frac{2}{\pi }\left(\frac{\frac{\pi }{2}t}{\tan \frac{\pi }{2}t}\right)=\frac{2}{\pi }$