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  • Evaluate the following: <img alt="\lim_{x\rightarrow 0}\left ( 1+\sin ^{2} x\right )^{\frac{1}{\left ( 1-cos^{2}x \right )}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim_%7Bx%5Crigh

Evaluate the following: \lim_{x\rightarrow 0}\left ( 1+\sin ^{2} x\right )^{\frac{1}{\left ( 1-cos^{2}x \right )}}

Option: 1

1


Option: 2

0


Option: 3

\infty


Option: 4

e


Answers (1)

best_answer

Given: \lim_{x\rightarrow 0}\left ( 1+\sin ^{2} x\right )^{\frac{1}{\left ( 1-cos^{2}x \right )}}

By using the trigonometric formula 1-\cos^{2}x= \sin ^{2}x,then above equation becomes,

\lim_{x\rightarrow 0}\left ( 1+\sin ^{2} x\right )^{\frac{1}{\left ( \sin^{2}x \right )}}
Using the property of the limit \lim_{x\rightarrow a}\left ( 1+ x\right )^{\frac{1}{x}}= e,

\lim_{x\rightarrow 0}\left ( 1+ \sin ^{2}x\right )^{\frac{1}{\left ( \sin ^{2}x \right )}}= e

Therefore, \lim_{x\rightarrow 0}\left ( 1+ \sin ^{2}x\right )^{\frac{1}{\left ( 1-\cos ^{2}x \right )}}= e.

Posted by

Sanket Gandhi

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