# Math Help - limit of (1+(1/x))^x

1. ## limit of (1+(1/x))^x

I am having trouble showing that $\lim_{x\to \infty}(1+\frac{1}{x})^x = e$.
All I have gotten is that I somehow have to use the sequence $e_{n}=(1+\frac{1}{n})^n$. Im not really sure where to start though.

2. One possible way is to valuate...

$\ln (1+\frac{1}{x})^{x} = x\cdot \ln (1+\frac{1}{x})= x \cdot (\frac{1}{x} - \frac{1}{2x^{2}} + \frac{1}{3x^{3}} - ...)$ (1)

From (1) is evident that...

$\lim_{x \rightarrow \infty} x\cdot \ln (1+\frac{1}{x}) =1$ (2)

Kind regards

$\chi$ $\sigma$

3. how could I prove it without using l'hopitals rule, i.e. using sequences