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  • Evaluate the limit of the following: <img alt="\lim _{x \rightarrow 0}[1+(1-\cos x)]^{1 / x^2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%200%7D%5B1&pl

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

Option: 1

\sqrt{e}


Option: 2

0


Option: 3

\frac{2}{n}


Option: 4

e


Answers (1)

best_answer

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

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{1-\cos x}{x^2}}

Use the trigonometric formula 1-\cos 2x= 2\sin ^{2}x, the equation becomes,

=e^{\lim _{x \rightarrow 0} \frac{2 \sin ^2 x / 2}{x^2}}
Multiply and divide by \frac{1}{2} , then the equation becomes,

=e^{\lim _{x \rightarrow 0}\left[\frac{\sin x / 2}{x / 2}\right] ^{2}\frac{1}{2}}
Use the limit formula \lim _{x \rightarrow 0} \frac{\sin x}{x}=1.
=e^{1 \cdot 1 / 2}
=e^{1 / 2}
=\sqrt{e}
Therefore,  \lim _{x \rightarrow 0}[1+(1-\cos x)]^{1 / x^2}=\sqrt{e}.

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