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