If $|x| < 1$,then $\lim_{n \to \infty} \{(1 + x)(1 + x^2)(1 + x^4) \dots (1 + x^{2^n})\}$ is equal to

$\lim _{x \rightarrow 0} \frac{e^x-e^{\sin x}}{2(x-\sin x)}$

If $f(x)$ satisfies $97 f(x) + m f\left(\frac{1}{x}\right) = 0$,where $f(x) = \lim_{n \rightarrow \infty} n(x^{1/n} - 1)$ for $x > 0$,then the value of $m$ is:

If $0 < p < q$,then $\lim _{n \rightarrow \infty}\left(q^n+p^n\right)^{1 / n}$ is equal to :

If a function $f$ is defined by $f(x) = \frac{\cot^3 x - \tan x}{\cos(x + \pi/4)}$ for $x \neq \pi/4$,then $\lim_{x \rightarrow \pi/4} f(x) = $

Let $f(x) = \frac{1}{3} x \sin x - (1 - \cos x)$. The smallest positive integer $k$ such that $\lim_{x \rightarrow 0} \frac{f(x)}{x^k} \neq 0$ is