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

 

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