Question

Let $f(n)=\displaystyle\lim _{x \rightarrow 0}\left(\left(1+\sin \frac{x}{2}\right)\left(1+\sin \frac{x}{2^{2}}\right) \ldots .\left(1+\sin \frac{x}{2^{n}}\right)\right)^{1 / x} .$ If $\displaystyle\lim _{n \rightarrow \infty} f(n)=\frac{3 e}{k}$ then find $k$

Answer 1

$f(n)=\displaystyle\lim _{x \rightarrow 0}\left(1+\sin \frac{x}{2}\right)^{1 / x}\left(1+\sin \frac{x}{2^{2}}\right)^{1 / x}$
$\cdots \cdots \cdots\left(1+\sin \frac{x}{2^{n}}\right)^{1 / x}$
$=e^{\frac{1}{2}} \cdot e^{\frac{1}{2^{2}}} \ldots \ldots . e^{\frac{1}{2^{n}}}$
Now, $\displaystyle\lim _{n \rightarrow \infty} f(n)=e^{\frac{1 / 2}{1-1 / 2}}=e$