Question

Evaluate the following:
$\underset{x\to 0}{lim}(1+\sin x{)}^{\cot x}$

Answer 1

$\underset{x\to 0}{lim}(1+\sin x{)}^{\cot x}=\underset{x\to 0}{lim}{e}^{\sin x\cdot \cot x}$ .....(using definition of $e^x$)
$=\underset{x\to 0}{lim}{e}^{\frac{\sin x\cdot cosx}{sinx}}$
$=e^{\underset{x\to 0}{lim}\cos x}$
$=e^{\cos 0}$
$=e$