Limx- 0sin - cos 2x -x2 equals

The correct option is B π lim_x→ 0sin ( π cos ^2x )/x^2=lim_→ 0sin ( π ( 1 sin ^2x ) )/x^2 =lim_x→ 0sin ( π sin ^2x )/x^2=πlim_x→ 0sin ( π s ...

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Standard XII

Mathematics

Algebra of Limits

limx→ 0sin π ...

Question

$lim x \to 0 \frac{sin (π {cos}^{2} x)}{x^{2}}$ equals

- π

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π

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\frac{π}{2}

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Solution

The correct option is B $π$
$lim x \to 0 \frac{sin (π {cos}^{2} x)}{x^{2}} = lim \to 0 \frac{sin (π (1 - {sin}^{2} x))}{x^{2}}$
$= lim x \to 0 \frac{sin (π {sin}^{2} x)}{x^{2}} = π lim x \to 0 \frac{sin (π {sin}^{2} x)}{π {sin}^{2} x} \times \frac{{sin}^{2}}{x^{2}}$
$= π \times 1 \times 1 = π$
Hence, option 'B' is correct.

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limx→0sin(πcos2x)x2 equals

The correct option is B πlimx→0sin(πcos2x)x2=lim→0sin(π(1−sin2x))x2=limx→0sin(πsin2x)x2=πlimx→0sin(πsin2x)πsin2x×sin2x2=π×1×1=πHence, option 'B' is correct.

$lim x \to 0 \frac{sin (π {cos}^{2} x)}{x^{2}}$ equals