Question
Evaluate $$\mathop {\lim }\limits_{x \to \infty } \,{2^{x - 1}}\tan \left( {\frac{a}{{{2^x}}}} \right)$$
A.
$$a$$
B.
$$2a$$
C.
$$\frac{a}{2}$$
D.
$$4a$$
Answer :
$$\frac{a}{2}$$
Solution :
$$\eqalign{
& {\text{We have,}} \cr
& \mathop {\lim }\limits_{x \to \infty } \,{2^{x - 1}}\tan \left( {\frac{a}{{{2^x}}}} \right) \cr
& = \mathop {\lim }\limits_{x \to \infty } \,\frac{a}{2}\frac{{\tan \left( {\frac{a}{{{2^x}}}} \right)}}{{\left( {\frac{a}{{{2^x}}}} \right)}}\,\,\,\left( {\frac{0}{0}{\text{ form}}} \right) \cr
& = \frac{a}{2}\mathop {\lim }\limits_{y \to 0} \frac{{\tan \,y}}{y} = \frac{a}{2}\,\,\,\,\,\left( {{\text{where }}y = \frac{a}{{{2^x}}}} \right) \cr} $$