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  • Evaluate the limit of the following: <img alt="\lim _{x \rightarrow 0}[1+(\sqrt{1+2 x}-1)]^{1 / x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%200%7D%5

Evaluate the limit of the following: \lim _{x \rightarrow 0}[1+(\sqrt{1+2 x}-1)]^{1 / x}

Option: 1

\sqrt{e}


Option: 2

e^{9/8}


Option: 3

\frac{2}{n}


Option: 4

e


Answers (1)

best_answer

Given: \lim _{x \rightarrow 0}[1+(\sqrt{1+2 x}-1)]^{1 / x}

If \lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} g(x)=0$, then $e^{\lim _{x \rightarrow a} \frac{f(x)}{g(x)}}.

=e^{\lim _{x \rightarrow 0} \frac{\sqrt{1+2x}-1}{x}}

=e^{\lim _{x \rightarrow 0} 2\left[\frac{\sqrt{1+2 x}-1}{(1+2 x)-1}\right]}

Apply the change of limit rule and take y= 1+2x.

=e^{2 \lim _{y \rightarrow 1}\left(\frac{\sqrt{y}-1}{y-1}\right)}
=e^{2(1 / 2)}
=e
Therefore, \lim _{x \rightarrow 0}[1+(\sqrt{1+2 x}-1)]^{1 / x}=e.



 

Posted by

HARSH KANKARIA

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