Finding the limit of function - exponential one Find the value of $\displaystyle \lim_{x \rightarrow 0}\left(\frac{1+5x^2}{1+3x^2}\right)^{\frac{1}{\large {x^2}}}$
We can write this limit function as : 
$$\lim_{x \rightarrow 0}\left(1+ \frac{2x^2}{1+3x^2}\right)^{\frac{1}{\large{x^2}}}$$
Please guide further how to proceed in such limit..

Let $n = \displaystyle\frac{1}{x^{2}}$:
$\displaystyle \lim_{x \rightarrow 0}\left(\frac{1+5x^2}{1+3x^2}\right)^{\frac{1}{\large {x^2}}}$ = $\displaystyle\lim_{n \rightarrow \infty}\left(\frac{n + 5}{n + 3}\right)^{n}$ = $\displaystyle\lim_{n \rightarrow \infty}\left(1 + \frac{2}{n + 3}\right)^{n}$ = $\displaystyle\lim_{n \rightarrow \infty}\left(1 + \frac{2}{n}\right)^{n}$ = $e^{2}$ 
by definition of $e$.
