Solution:
\( \lim \limits _{x \rightarrow \infty} \left ( 1+ \frac{1}{x} \right ) ^x \)
\( = e^{\ln \left ( \lim \limits _{x \rightarrow \infty} \left ( 1+ \frac{1}{x} \right ) ^x \right ) } \)
\( = e^{ \lim \limits _{x \rightarrow \infty} \ln \left ( 1+ \frac{1}{x} \right ) ^x } \)
\( = e^{ \lim \limits _{x \rightarrow \infty}x \ln \left ( 1+ \frac{1}{x} \right ) } \)
\( = e^{ \lim \limits _{x \rightarrow \infty}\frac{\ln \left ( 1+ \frac{1}{x} \right )}{\frac{1}{x}} }\)
\( = e^{ \lim \limits _{x \rightarrow \infty} \frac{\frac{1}{\left ( 1 + \frac{1}{x} \right )}\frac{-1}{x^2}}{\frac{-1}{x^2}} }\) by LHR
\( = e^{ \lim \limits _{x \rightarrow \infty} \frac{1}{1+\frac{1}{x}}} \)
\( = e^1 = e \)
