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  • Evaluating: <img alt="\mathrm{ \lim _{x \rightarrow 0}\left(\frac{\tan \left(\pi \cos ^2 x\right)}{x^2}\right) }" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%20%5Clim%2

Evaluating:
\mathrm{ \lim _{x \rightarrow 0}\left(\frac{\tan \left(\pi \cos ^2 x\right)}{x^2}\right) }

Option: 1

0


Option: 2

4


Option: 3

5


Option: 4

1


Answers (1)

best_answer

Depends on what you're willling to take

\mathrm{\text { Heuristically } \cos (x) \sim 1-\frac{1}{2} x^2 \text { and so } \cos (x)^2 \sim 1-x^2 \text {. Thus, if } L \text { denotes your limit: }}

                                                 \mathrm{L=\lim _{x \rightarrow 0} \frac{\tan \left(\pi-\pi x^2\right)}{x^2}=-\lim _{x \rightarrow 0} \frac{\tan \left(\pi x^2\right)}{x^2}=-\pi}

where the last equality follows from substitution \mathrm{ \pi x^2 \mapsto t}  and the (common) fact that

                                                              \mathrm{\lim _{t \rightarrow 0} \frac{\tan (t)}{t}=1}

Posted by

seema garhwal

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