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Question
\[\lim_{x \to \infty} a^x \sin \left( \frac{b}{a^x} \right), a, b > 1\] is equal to
Options
b
a
a loge b
b loge a
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Solution
b
\[\lim_{x \to \infty} a^x \sin \left( \frac{b}{a^x} \right)\]
\[ \lim_{x \to \infty} b\left( \frac{\sin \frac{b}{a^x}}{\frac{b}{a^x}} \right)\]
\[Let \frac{b}{a^x} = y\]
\[x \to \infty \]
\[ \therefore y \to 0\]
\[ \lim_{y \to 0} \frac{b \sin y}{y} = b\]
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