# What is the limit of (1+2x)^(1/x) as x approaches infinity?

Aug 8, 2016

1

#### Explanation:

${\lim}_{x \to \infty} {\left(1 + 2 x\right)}^{\frac{1}{x}}$

$= {\lim}_{x \to \infty} {e}^{\ln {\left(1 + 2 x\right)}^{\frac{1}{x}}}$

$= {e}^{{\lim}_{x \to \infty} \ln {\left(1 + 2 x\right)}^{\frac{1}{x}}}$ as exponential function is continous

$= {e}^{L}$

$L = {\lim}_{x \to \infty} \ln {\left(1 + 2 x\right)}^{\frac{1}{x}}$

$= {\lim}_{x \to \infty} \frac{1}{x} \ln \left(1 + 2 x\right)$

$= {\lim}_{x \to \infty} \frac{\ln \left(1 + 2 x\right)}{x}$

which is $\frac{\infty}{\infty}$ indeterminate so we can use L Hopital

$= {\lim}_{x \to \infty} \frac{\frac{1}{1 + 2 x}}{1}$

$\implies L = {\lim}_{x \to \infty} \frac{1}{1 + 2 x} = 0$

and
${e}^{0} = 1$