I do understand that $\displaystyle e$ is the number such that

$\displaystyle e=\lim_{n \rightarrow \infty} \left( 1+\frac{1}{n} \right)^n$

and how it is obtained.

But I don't understand how the definition of the function $\displaystyle e^x$ as a limit of a sequence is obtained, i.e.

$\displaystyle e^x=\lim_{n\rightarrow\infty}\left( 1+\frac{x}{n} \right)^n$

Is there a proof?

Appreciate those who help. Thanks!