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

Evaluate   \mathrm{\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x-\sin x}}

Option: 1

0


Option: 2

-1


Option: 3

1


Option: 4

2


Answers (1)

best_answer

\mathrm{\lim _{x \rightarrow 0}\left(\frac{1}{\sin (x)}-\frac{1}{x}\right) =\lim _{x \rightarrow 0}\left(\frac{x-\sin (x)}{x \sin (x)}\right)}
                                        \mathrm{=\lim _{x \rightarrow 0}\left(\frac{x-\left(x-\frac{x^{3}}{6}+O\left(x^{5}\right)\right)}{x\left(x-\frac{x^{3}}{6}+O\left(x^{5}\right)\right)}\right) }
                                         \mathrm{=\lim _{x \rightarrow 0}\left(\frac{\frac{x^{3}}{6}-O\left(x^{5}\right)}{x^{2}-\frac{x^{4}}{6}+O\left(x^{6}\right)}\right) }
                                         \mathrm{=\lim _{x \rightarrow 0}\left(\frac{\frac{x}{6}-O\left(x^{3}\right)}{1-\frac{x^{2}}{6}+O\left(x^{4}\right)}\right) }
                                         \mathrm{=\frac{0}{1} }
                                         \mathrm{ =0 } 

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Gaurav

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