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Question
\[\lim_{x \to \infty} \frac{\sin x}{x}\] equals
Options
0
∞
1
does not exist
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Solution
(a) 0
\[\lim_{x \to \infty} \frac{\sin x}{x}\]
\[Let x = \frac{1}{y}\]
\[ x \to \infty \]
\[ \therefore y \to 0\]
\[ = \lim_{y \to 0} \frac{\sin \frac{1}{y}}{\frac{1}{y}}\]
\[ = \lim_{y \to 0} y \sin \frac{1}{y}\]
\[ = \lim_{y \to 0} y \times \lim_{y \to 0} \sin \frac{1}{y}\]
\[ = 0 \times \lim_{y \to 0} \sin \frac{1}{y}\]
\[ = 0\]
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