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

Answer

Correct option: C.
$4 \pi^{2}$
c
$\lim _{x \rightarrow 0} \frac{\sin ^{2}\left(\pi \cos ^{4} x\right)}{x^{4}}$

$\lim _{x \rightarrow 0} \frac{1-\cos \left(2 \pi \cos ^{4} x\right)}{2 x^{4}}$

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

$=\frac{1}{2} \cdot 4 \pi^{2} \cdot \frac{1}{2}(2)^{2}=4 \pi^{2}$

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