Let f(x)= cl x- x x- 4 , & x 4 a, & x= 4 . The value of a so that… - Vidyadip

MCQ

Let $f(x)=\left\{\begin{array}{cl}\frac{\tan x-\cot x}{x-\frac{\pi}{4}}, & x \neq \frac{\pi}{4} \\ \text { a, } & x=\frac{\pi}{4}\end{array}\right.$ The value of a so that $f(x)$ is continuous , $x=\frac{\pi}{4}$, is

A
2
✓
4
C
3
D
1

✓

Answer

Correct option: B.

4

(B)
Since $f (x)$ is continuous at $x=\frac{\pi}{4}$.
$\therefore f \left(\frac{\pi}{4}\right)=\lim _{x \rightarrow \frac{\pi}{4}} f (x)$
$\Rightarrow a =\lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan x-\cot x}{x-\frac{\pi}{4}}$
Applying L'Hospital rule on R.H.S., we get
$a =\lim _{x \rightarrow \frac{\pi}{4}} \frac{\sec ^2 x+\operatorname{cosec}^2 x}{1}$
$\Rightarrow a =(\sqrt{2})^2+(\sqrt{2})^2=4$

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