Question

Solve $\underset{x\to \pi /2}{lim}{\left(\frac{1+\cot x}{1+\cos x}\right)}^{1/\cos x}$

Answer 1

Let, $L=\underset{x\to 0}{lim}{\left(\frac{1+\cot x}{1+\cos x}\right)}^{1/\cos x}$
Clearly form of the limit is $1^{\infty }$, So
$L=\underset{x\to 0}{lim}{(1+\frac{1+\cot x}{1+\cos x}-1)}^{1/\cos x}$
$=\underset{x\to 0}{lim}{(1+\frac{\cot x-\cos x}{1+\cos x})}^{\frac{1+\cos x}{\cot x-\cos x}.\frac{\cot x-\cos x}{(1+\cos x)\cos x}}$
$=e^{\underset{x\to 0}{lim}\frac{\cot x-\cos x}{(1+\cos x)\cos x}}$
$=e^{\underset{x\to 0}{lim}\frac{\frac{1}{\sin x}-1}{(1+\cos x)}}=e^0=1$