Evaluate $\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{f(x)/x}$

This is a very good point knowing that if $$\lim_{x\to{+\infty}} f(x)^{g(x)}=1^{+\infty}$$ which is indeterminate limit then we can solve it by taking the following limit: $$k =\lim_{x\to +\infty}\big(f(x)-1\big)g(x)$$ instead. So $$\lim_{x\to{+\infty}} f(x)^{g(x)}=e^k$$

Now, try to use this formula also. It gives you $~~\text{e}~~$ at last. For more see how @Brian proved me that. This proof deserves more that +100. 100