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  • Evaluate <img alt="\mathrm{\lim _{x \rightarrow 0^{+}} \frac{\sin ^2 x \tan x-x^3}{x^7}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bx%20%5Crightarrow%200%5E

Evaluate \mathrm{\lim _{x \rightarrow 0^{+}} \frac{\sin ^2 x \tan x-x^3}{x^7}}

Option: 1

\frac{1}{15}


Option: 2

0


Option: 3

\frac{2}{3}


Option: 4

1


Answers (1)

best_answer

L'Hospital once the fraction becomes

                                     \mathrm{ \frac{2 \sin ^2 x+\tan ^2 x-3 x^2}{7 x^6} }
Still an indeterminate

L'hospital 6 more times and rearranging the fraction becomes

                                    \mathrm{ \frac{4 \cos 2 x}{315}+\sec ^8 x-\frac{4 \sec ^6 x}{3}+\frac{2 \sec ^4 x}{5}-\frac{4 \sec ^2 x}{315} }

And the limit for \mathrm{x \rightarrow 0} is \mathrm{1-\frac{4}{3}+\frac{2}{5}=\frac{1}{15}}

Posted by

Anam Khan

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