Question

$\underset{n\to \infty }{lim}{(1+\sin \left(\frac{a}{n}\right))}^n=$

• A. $e$
• B. $e^a$
• C. $a^e$
• D. $a$

Answer 1

The correct option is B $e^a$

$\underset{n\to \infty }{lim}{(1+\sin \left(\frac{a}{n}\right))}^n,\left[1^{\infty }\text{form}\right]$
$=e^{\underset{n\to \infty }{lim}n(1+\sin \left(\frac{a}{n}\right)-1)}$
$=e^{\underset{n\to \infty }{lim}n\left(\sin \left(\frac{a}{n}\right)\right)}$
Multiplying and dividing by $a$
$=e^{\underset{n\to \infty }{lim}\sin \left(\frac{a}{n}\right)\times \frac{a}{a/n}}$
$=e^a$