TITLE: Problem evaluating limits with the variable in the exponent
QUESTION [7 upvotes]: I have problem evaluating limits with the variable in power, like the following limits:

$\lim_{x \to 0} (1+ \sin 2x)^{\frac{1}{x}}$
$\lim_{x \to \infty} \big(\frac{2x+5}{2x-1})^{2x}$

I asked the question like this to get the main idea behind evaluating these kind of limits, so I can solve all the related questions.

REPLY [7 votes]: A good idea whenever the exponent is variable is to take logarithims. So for example, to compute the first limit first find the limit of $$ \log \left( 1+\sin 2x 
\right)^{1/x} = \frac{1}{x} \cdot \log (1+ \sin 2x)  .$$ As $x\to 0$ this is of the form $0/0$ so by applying L'Hopitals rule we find that the limit is equal to $ \lim_{x\to 0} \frac{ 2\cos 2x}{1+\sin 2x} = 2$, which in turn implies the original limit is $e^2.$