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Question
\[\lim_{n \to \infty} \frac{\sin \left( \frac{a}{2^n} \right)}{\sin \left( \frac{b}{2^n} \right)}\]
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Solution
\[\lim_{n \to \infty} \frac{\sin \left( \frac{a}{2^n} \right)}{\sin \left( \frac{b}{2^n} \right)}\]
\[ = \lim_{n \to \infty} \frac{\left( \frac{a}{2^n} \right) \sin \left( \frac{a}{2^n} \right)}{\left( \frac{a}{2^n} \right) \times \left( \frac{b}{2^n} \right) \times \left( \frac{\sin \left( \frac{b}{2^n} \right)}{\frac{b}{2^n}} \right)}\]
\[\text{ Let }: \]
\[ y = \frac{a}{2^n}\]
\[ z = \frac{b}{2^n}\]
\[If n \to \infty , then y \to 0 and z \to 0 . \]
\[ = \frac{y}{z} \lim_{y \to 0} \frac{\sin y}{y} \times \frac{1}{\lim_{z \to 0} \frac{\sin z}{z}}\]
\[ = \frac{y}{z} \times 1 \times \frac{1}{1}\]
\[ = \frac{\frac{a}{2^n} \times 1}{\frac{b}{2^n} \times 1} \]
\[ = \frac{a}{b}\]
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