Evaluate the following limit : lim(x→∞) 2^(n - 1) sin (a/2^n)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to\infty}2^{n-1}sin(\cfrac{a}{2^n}) \)

lim(x→∞) 2^{(n - 1)} sin (a/2ⁿ)

Answer 1

We have \(\lim\limits_{\text x \to\infty}2^{n-1}sin(\cfrac{a}{2^n}) \)

We know, \(\lim\limits_{\theta \to0}\cfrac{sin\,\theta}{\theta} \) = 1 then, we get

\(\lim\limits_{\text x \to\infty}2^{n-1}sin(\cfrac{a}{2^n}) \) = \(\cfrac a2\)

Hence, \(\lim\limits_{\text x \to\infty}2^{n-1}sin(\cfrac{a}{2^n}) \) = \(\cfrac a2\)