The value of $\underset{x\to a}{lim} {\left(\frac{\sin x}{\sin a}\right)}^{\frac{1}{x-a}}$ , is

Solution:

$\begin{array}{l}\underset{x\to a}{ lim} {\left(\frac{\sin x}{\sin a}\right)}^{\frac{1}{x-a}}\\ =\underset{x\to a}{lim} {\left\{1+\frac{\sin x-\sin a}{\sin a}\right\}}^{\frac{1}{x-a}}\\ =e^{\underset{x\to a}{lim} \frac{\sin x-\sin a}{x-a}\times \frac{1}{\sin a}}=e^{\frac{\cos a}{\sin a}}=e^{\cot a}\end{array}$