If $f(x)=x^{100}+x^{99}+\ldots+x+1$, then $f^{\prime}(1)$ is equal to
If $f(x)=1-x+x^2-x^3+\ldots-x^{99}+x^{100}$, then $f^{\prime}(1)$ is equal to
If $f(x)=\frac{\tan x}{x-\pi}$, then $\lim _\limits{x \rightarrow \pi} f(x)=$
Given, $\begin{aligned} f(x)=\frac{\tan x}{x-\pi} & =\lim _{x \rightarrow \pi} \frac{\tan x}{x-\pi}=\lim _{\pi-x \rightarrow 0} \frac{-\tan (\pi-x)}{-(\pi-x)} \quad\left[\because \lim _{x \rightarrow 0} \frac{\tan x}{x}=1\right] \\ & =1\end{aligned}$
$\lim _\limits{x \rightarrow 0}\left(\sin m x \cot \frac{x}{\sqrt{3}}\right)=2$, then $m=$ ...........
$$\begin{aligned} &\text { Given, } \lim _{x \rightarrow 0}\left(\sin m x \cot \frac{x}{\sqrt{3}}\right)=2\\ &\begin{aligned} & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot m x \cdot \frac{1}{\tan \frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot m x \cdot \frac{\frac{x}{\sqrt{3}}}{\tan \frac{x}{\sqrt{3}}} \cdot \frac{\frac{1}{x}}{\frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot \lim _{x \rightarrow 0} \frac{\frac{x}{\sqrt{3}}}{\tan \frac{x}{\sqrt{3}}} \cdot \lim _{x \rightarrow 0} \frac{m x}{\frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \sqrt{3} x=2 \\ \therefore\quad & m=\frac{2 \sqrt{3}}{3} \end{aligned} \end{aligned}$$
If $y=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots$, then $\frac{d y}{d x}=$ .........
$$\begin{aligned} \text{Given,}\quad y & =1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots \\ \therefore\quad \frac{d y}{d x} & =0+1+\frac{2 x}{2}+\frac{3 x^2}{6}+\frac{4 x^3}{4!} \\ & =1+x+\frac{x^2}{2}+\frac{x^3}{6}+\ldots \\ & =1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots \\ & =y \end{aligned}$$