Question

$\displaystyle\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$ is equal to :

• A. $\pi^{2}$
• B. $2 \pi^{2}$
• C. $4 \pi^{2}$
• D. $4 \pi$

Answer 1

Answer: (C)

$\displaystyle\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$
$\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos \left(2 \pi \cos ^{4} x\right)}{2 x^{4}}$
$\displaystyle\lim _{x \rightarrow 0} \frac{1-\cos \left(2 \pi-2 \pi \cos ^{4} x\right)}{\left[2 \pi\left(1-\cos ^{4} x\right)\right]^{2}} 4 \pi^{2} \cdot \frac{\sin ^{4} x}{2 x^{4}}\left(1+\cos ^{2} x\right)^{2}$
$=\frac{1}{2} \cdot 4 \pi^{2} \cdot \frac{1}{2}(2)^{2}=4 \pi^{2}$