# Limit of Functions in Indeterminate Form Example

#### Find $\lim _{x\to 0}\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}$.

SOLUTION:

$\lim _{x\to 0}\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}$

Substitute $x=0$

$=\left(1+0\right)^{\frac{1}{0}}$

$1^{\infty }$

This is in indeterminate form.

Let $N=\lim _{x\to 0}\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}$

$ln\:N=ln\:\lim _{x\to 0}\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}$

$ln\:N=\lim _{x\to 0}\:ln\:\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}$

$ln\:N=\lim _{x\to 0}\:\frac{1}{\sqrt{x}}\:ln\:\left(1+2\sqrt{x}\right)$

$ln\:N=\frac{0}{0}$

This is indeterminate form, we apply L’Hospital’s Rule

$ln\:N=\lim _{x\to 0}\:\left(\frac{\frac{1}{1+2\sqrt{x}}\cdot x^{-\frac{1}{2}}}{\frac{1}{2}x^{-\frac{1}{2}}}\right)$

$ln\:N=\lim _{x\to 0}\:\frac{2}{1+2\sqrt{x}}$

$ln\:N=\frac{2}{1+0}$

$ln\:N=2$

$N=e^2$

Therefore,

$\lim _{x\to 0}\left(1+2\sqrt{x}\right)^{\frac{1}{\sqrt{x}}}=e^2$