How to calculate $\lim \limits_{n \to \infty}(1 +\frac{a}{n})^n$?

I know what solution is $e^a$ but I don't know how to calculate this limit:

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

Can someone explain the steps to me?

Solution 1:

$$\lim \limits_{x \to +\infty} (1+a/x)^x = e^{\lim \limits_{x \to +\infty} x \log (1+a/x)}$$

Then apply l'Hôpital's rule to $\frac{\log (1+a/x)}{1/x}$ to get $$\frac{\frac{1}{1+a/x} \cdot \frac{-a}{x^2}}{\frac{-1}{x^2}} = \frac{a}{1+a/x} = a$$

Solution 2:

Use $\lim_{n\rightarrow\infty} (1+1/n)^n=e$.

Let $m=n/a$ then $\lim_{n\rightarrow\infty} (1+a/n)^n=\lim_{m\rightarrow \infty} (1+1/m)^{ma}=e^a$.