limit of product of $(a_1a_2.\dots a_n)^{\frac{1}{n}}$ How to calculate the following limit 
$$ \lim_{n\rightarrow \infty} \left[ \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)\cdots \left(1+\frac{n}{n}\right) \right]^\frac{1}{n} .$$
 I was trying this by taking the $\log $ of the product and then limit but I am not getting the answer could anybody please help me. And, also is there any general rule for calculating the limit for  $$ (a_1a_2.\dots a_n)^{\frac{1}{n}}.$$
Thanks. 

This answer uses Sterling's approximation to get the limit:
\begin{align}
l=\;&\left[ \left( 1+\frac { 1 }{ n }  \right) \left( 1+\frac { 2 }{ n }  \right) \cdots \left( 1+\frac { n }{ n }  \right)  \right]^{ \frac { 1 }{ n } } \\
=\; & \left[ \frac{1}{n^n}(n+1)(n+2)\dots(n+n) \right]^{ \frac { 1 }{ n } } \\
= \;& \frac{1}{n}\left[ \frac{(2n)!}{n!} \right]^{ \frac { 1 }{ n } }  \\
\approx\; & \frac{1}{n}\left[\sqrt{2}\cdot  2^{2n}\left(\frac{n}{e} \right)^n\right]^{\frac{1}{n}} \Rightarrow \\
\lim_{n\rightarrow \infty}l=  \;& \frac{4}{e}
\end{align}
