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Question

Question: \(\lim_{x \rightarrow 0}\frac{\sin\left( \pi\cos^{2}x \right)}{x^{2}}\) =...

lim⁡x→0sin⁡(πcos⁡2x)x2\lim_{x \rightarrow 0}\frac{\sin\left( \pi\cos^{2}x \right)}{x^{2}} =

A

−π

B

π

C

π/2

D

1

Answer

π

Explanation

Solution

lim⁡x→0sin⁡(πcos⁡2x)x2\lim_{x \rightarrow 0}\frac{\sin\left( \pi\cos^{2}x \right)}{x^{2}}= lim⁡x→0sin⁡(π−πsin⁡2x)x2\lim_{x \rightarrow 0}\frac{\sin\left( \pi - \pi\sin^{2}x \right)}{x^{2}}

[sin⁡(π−θ=sin⁡θ)]\left\lbrack \sin\left( \pi - \theta = \sin\theta \right) \right\rbrack

= lim⁡x→0sin⁡(πsin⁡2x)πsin⁡2x\lim_{x \rightarrow 0}\frac{\sin\left( \pi\sin^{2}x \right)}{\pi\sin^{2}x}× (πsin⁡2x)x2\frac{\left( \pi\sin^{2}x \right)}{x^{2}}= π