73
MCQ (Single Correct Answer)
If $f(x)=1+x+\frac{x^2}{2}+\ldots+\frac{x^{100}}{100}$, then $f^{\prime}(1)$ is equal to
A
$\frac{1}{100}$
B
100
C
0
D
Does not exist
74
MCQ (Single Correct Answer)
If $f(x)=\frac{x^n-a^n}{x-a}$ for some constant $a$, then $f^{\prime}(a)$ is equal to
A
1
B
0
C
$\frac{1}{2}$
D
Does not exist
75
MCQ (Single Correct Answer)
If $f(x)=x^{100}+x^{99}+\ldots+x+1$, then $f^{\prime}(1)$ is equal to
A
5050
B
5049
C
5051
D
50051
76
MCQ (Single Correct Answer)
If $f(x)=1-x+x^2-x^3+\ldots-x^{99}+x^{100}$, then $f^{\prime}(1)$ is equal to
A
150
B
$-$50
C
$-$150
D
50
77
If $f(x)=\frac{\tan x}{x-\pi}$, then $\lim _\limits{x \rightarrow \pi} f(x)=$
Explanation
Given, $\begin{aligned} f(x)=\frac{\tan x}{x-\pi} & =\lim _{x \rightarrow \pi} \frac{\tan x}{x-\pi}=\lim _{\pi-x \rightarrow 0} \frac{-\tan (\pi-x)}{-(\pi-x)} \quad\left[\because \lim _{x \rightarrow 0} \frac{\tan x}{x}=1\right] \\ & =1\end{aligned}$