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Existence of Limit

Evaluate the ...

Question

Evaluate the following limit :
$lim x \to 0 \frac{{sin}^{2} 3 x}{x^{2}}$

A

1

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B

3

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C

9

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D

0

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Solution

The correct option is A 9
${lim}_{x \to 0} \frac{{sin}^{2} 3 x}{x^{2}}$

$= {lim}_{x \to 0} \frac{sin 3 x}{x} \times \frac{sin 3 x}{x}$

$= {lim}_{x \to 0} 3 (\frac{sin 3 x}{3 x}) \times 3 (\frac{sin 3 x}{3 x})$

$= 3 {lim}_{x \to 0} \frac{sin 3 x}{3 x} \times 3 {lim}_{x \to 0} \frac{sin 3 x}{3 x} = 3 \times 3 = 9$

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