Question

$\underset{n\to \infty }{\lim }{2}^{n-1}\sin \left(\frac{a}{2^n}\right)$

Answer 1

$$\begin{gathered} \underset{n\to \infty }{\lim }{2}^{n-1}\sin \left(\frac{a}{2^n}\right) \\ =\underset{n\to \infty }{\lim }\frac{2^n}{2}\times \frac{\sin \left({\displaystyle \frac{a}{2^n}}\right)}{\left({\displaystyle \frac{a}{2^n}}\right)}\times \left(\frac{a}{2^n}\right) \\ \mathrm{Let}y=\frac{a}{2^n} \\ \mathrm{If}n\to \infty ,\mathrm{then}y\to 0. \\ =\underset{y\to 0}{\lim }\frac{a}{2}\times \left(\frac{\sin y}{y}\right) \\ \Rightarrow \frac{a}{2} \end{gathered}$$