Question

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

• A. $e^{cota}$
• B. $e^{tana}$
• C. $e^{sina}$
• D. $e^{cosa}$

Answer 1

The correct option is A $e^{cota}$

$\underset{x\to a}{lim}{\left(\frac{\sin x}{\sin a}\right)}^{\frac{1}{x-a}}=\underset{x\to a}{lim}{(1+\frac{\sin x}{\sin a}-1)}^{\frac{1}{x-a}}$
$=\underset{x\to a}{lim}{(1+\frac{sinx-\sin a}{\sin a})}^{\frac{\sin a}{sinx-\sin a}\frac{1}{x-a}.\frac{sinx-\sin a}{\sin a}}$
$=e^{\underset{x\to a}{lim}\frac{1}{\sin a}.\frac{\sin x-\sin a}{x-a}}$
Using L-Hospital's rule,
$=e^{\underset{x\to a}{lim}\frac{1}{\sin a}.\frac{\cos x}{1}}=e^{\cot a}$