If a function of defined by f(x)= cc 1-2 x -4 x , & if x 4 , & if… - Vidyadip

MCQ

If a function of defined by
$f(x)=\left\{\begin{array}{cc}\frac{1-\sqrt{2} \sin x}{\pi-4 x}, & \text { if } x \neq \frac{\pi}{4} \\k, & \text { if } x=\frac{\pi}{4} \end{array}\right.$
is continuous at $x=\frac{\pi}{4}$, then $k =$

✓
$\frac{1}{4}$
B
1
C
$-\frac{1}{4}$
D
2

✓

Answer

Correct option: A.

$\frac{1}{4}$

(A)
Since $f (x)$ is continuous at $x=\frac{\pi}{4}$,
$\therefore \quad f\left(\frac{\pi}{4}\right)=\lim _{x \rightarrow \frac{\pi}{4}} f(x)$
$\Rightarrow k =\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\sqrt{2} \sin x}{\pi-4 x}$
Applying L'Hospital rule on R.H.S., we get
$k =\lim _{x \rightarrow \frac{\pi}{4}} \frac{-\sqrt{2} \cos x}{-4} \quad \Rightarrow k =\frac{1}{4}$

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