Limit of [1 + sin(x)]^(1/x) when x approaches 0

[rene]

Can somebody solve this problem: the limit of [1 + sin(x)]^(1/x) when x approaches 0 ?

 

[awvvu]

Take the logarithm of that expression, which will let you use L'Hopital to solve it.

 

[arildno]

The simplest is to rewrite this as:
$$(1+\sin(x))^{\frac{1}{x}}=((1+\sin(x))^{\frac{1}{\sin(x)}})^{\frac{\sin(x)}{x}}$$
The correct limit is quite easy to deduce from this.