Limit of (1+x)^(1/x)

To evaluate  
\[lim_{x \rightarrow \infty} (1+x)^{ \frac{1}{x}}\]
  take logs.
\[ln((1+x)^{ \frac{1}{x}})= \frac{1}{x} ln(1+x)= \frac{ln(1+x)}{x}\]
. This is of Indeterminate Form as  
\[x \rightarrow \infty\]
  (it equals  
\[\frac{ \infty}{ \infty }\]
)so use l'Hospital's Rule.
\[\begin{equation} \begin{aligned} lim_{x \rightarrow \infty} \frac{ln(1+x)}{x} &= lim_{ x \rightarrow \infty} \frac{\frac{d}{dx} (ln(1+x))}{\frac{d}{dx}(x)} \\ &= lim_{ x \rightarrow \infty} \frac{1/(1+x)}{1} \\ &= 0 \end{aligned} \end{equation}\]
.
Then  
\[lim_{x \rightarrow \infty} (1+x)^{ \frac{1}{x}}=e^0=1\]
.

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