\lim_{n\rightarrow \infty}\left(1+\frac x n\right)^n=\\
=\lim_{n\rightarrow \infty}\left[1+\frac x {1!}+\frac {x^2}{2!}\left(1-\frac 1 n\right)+\frac {x^3}{3!}\left(1-\frac 1 n\right)\left(1-\frac 2 n\right)+ ... +\frac {x^n}{n!}\left(1-\frac 1 n\right)\left(1-\frac 2 n\right) ... \left(1-\frac {n-2} n\right)\left(1-\frac {n-1} n\right)\right]=\\
=1+\frac {x}{1!} + \frac {x^2}{2!}\lim_{n\rightarrow \infty}\left(1-\frac 1 n\right)+\frac {x^3}{3!}\lim_{n\rightarrow \infty}\left[\left(1-\frac 1 n\right)\left(1-\frac 2 n\right)\right]+ ... +\lim_{n\rightarrow \infty}\left(\frac {x^n}{n!}\right)\lim_{n\rightarrow \infty}\left[\left(1-\frac 1 n\right)\left(1-\frac 2 n\right) ... \left(1-\frac {n-2} n\right)\left(1-\frac {n-1} n\right)\right]