Question

${lim}_{x\to a}{\left(\frac{\sin x}{\sin a}\right)}^{\frac{1}{x-a}},a\ne n\pi$, $n$ is an integer, equals

• A. $e^{\cot a}$
• B. $e^{\tan a}$
• C. $e^{\sin a}$
• D. $e^{\cos a}$

Answer 1

The correct option is A $e^{\cot a}$

limit $=e^L$

$L=\underset{x\to a}{lim}(\frac{\sin x}{\sin a}-1)\left(\frac{1}{x-a}\right)$

$=\underset{x\to a}{lim}\frac{\sin x-\sin a}{\sin a}\frac{1}{x-a}$

${\int }_{x\to a}\frac{2\cos \frac{x+a}{2}\sin \frac{x-a}{2}}{\left(sina\right)\frac{x-a}{2}2}$

$L=\frac{\cos a}{\sin a}-\cot a$

limit $=e^{cota}$