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Question
\[\lim_{x \to 0} \frac{\sin^2 4 x^2}{x^4}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{\sin^2 4 x^2}{x^4} \right]\]
\[= \lim_{x \to 0} \left[ \frac{\sin\left( 4 x^2 \right)}{x^2} \times \frac{\sin\left( 4 x^2 \right)}{x^2} \right]\]
\[ = \lim_{x \to 0} \left[ \frac{\sin\left( 4 x^2 \right)}{4 x^2} \times 4 \times \frac{\sin\left( 4 x^2 \right)}{4 x^2} \times 4 \right]\]
\[ = 4 \times 4 \left[ \because \lim_{x \to 0} \frac{\sin x}{x} = 1 \right]\]
\[ = 16\]
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