Question

Evaluate the following limits:
1. $\underset{x⟶\pi /2}{lim}(\frac{\pi }{2}-x)tanx$

Answer 1

$1.\underset{x\to \pi /2}{lim}(\frac{\pi }{2}-x)\tan x=\underset{x\to \pi /2}{lim}\frac{\frac{\pi }{2}-x}{\cot x}=$indeterminate form

Applying L'Hospital Rule

$\underset{x\to \pi /2}{lim}\frac{-1}{-{\text{cosec}}^{2}x}=\underset{x\to \pi /2}{lim}{\sin }^{2}x=1$