Question

The value of ${lim}_{n\to \infty }(1+sin\frac{a}{n}{)}^n$ is equal to

• A. 1
• B. $e^a$
• C. $e^{2a}$
• D. $e^{-3}$

Answer 1

The correct option is B $e^a$

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

As ${lim}_{n⟶\infty }\sin \frac{a}{n}={lim}_{n⟶\infty }\left(\frac{a}{n}\right)\frac{\sin \frac{a}{n}}{\frac{a}{n}}={lim}_{n⟶\infty }\frac{a}{n}$

So, ${lim}_{n⟶\infty }{(1+\sin \frac{a}{n})}^n={lim}_{n⟶\infty }{(1+\frac{a}{n})}^n$

Since, now undetermined form is of type $1^{\infty }$

So, ${lim}_{n⟶\infty }{(1+\frac{a}{n})}^n=e^{{lim}_{n⟶\infty }{(1+\frac{a}{n})}^n}=e^a$