Question

$\underset{n\to \infty }{lim}{(1+\sin \frac{a}{n})}^n$ is equal to

• A. $e^{a/2}$
• B. $e^a$
• C. $e$
• D. $e^{2a}$

Answer 1

The correct option is B $e^a$

${lim}_{n\to \infty }(1+sin\frac{a}{n}{)}^n$

Take $\frac{1}{n}=x$

As $n\to \infty ,x\to 0$

$\Rightarrow {lim}_{x\to 0}(1+sinax{)}^{\frac{1}{x}}$

This is in the form of $1^{\infty }$

We know that ${lim}_{x\to a}{f}^g=e^{{lim}_{x\to }(f-1)g}$

Here $f=1+sinax$ and $g=\frac{1}{x}$

$\Rightarrow e^{{lim}_{x\to 0}(1+sinax-1)\frac{1}{x}}$

$\Rightarrow e^{{lim}_{x\to 0}\left(sinax\right)\frac{a}{ax}}$

$\Rightarrow e^a$ (since${lim}_{x\to 0}\frac{sinx}{x}=1$)