limx→0 sin⁡πcos2⁡xx2 equals

# $\underset{x\to 0}{lim} \frac{\mathrm{sin}\left(\pi {\mathrm{cos}}^{2}x\right)}{{x}^{2}}$ equals

1. A

$-\pi$

2. B

$\mathrm{\pi }$

3. C

$\frac{\pi }{2}$

4. D

1

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### Solution:

We have,

$\underset{x\to 0}{lim} \frac{\mathrm{sin}\left(\pi {\mathrm{cos}}^{2}x\right)}{{x}^{2}}$

$\begin{array}{l}=\underset{x\to 0}{lim} \frac{\mathrm{sin}\left(\pi -\pi {\mathrm{cos}}^{2}x\right)}{{x}^{2}}\frac{x}{{x}^{2}}\\ =\underset{x\to 0}{lim} \frac{\mathrm{sin}\left(\pi {\mathrm{sin}}^{2}x\right)}{\pi {\mathrm{sin}}^{2}x}×\frac{\pi {\mathrm{sin}}^{2}x}{{x}^{2}}=1×\pi ×\left(1{\right)}^{2}=\pi \end{array}$

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