The value of k for which the function f(x)= cll 1- 4 x 8 x^2 , & … - Vidyadip

MCQ

The value of $k$ for which the function $f(x)=\left\{\begin{array}{cll}\frac{1-\cos 4 x}{8 x^2}, & \text { if } & x \neq 0 \\ k, & \text { if } & x=0\end{array}\right.$ is continuous at $x=0$ is

✓
$0$
B
$-1$
C
$1$
D
$2$

✓

Answer

Correct option: A.

$0$

Given, the function $f$ is continuous at $x=0$.
$\therefore \lim _{x \rightarrow 0} f(x)=f(0)$.
Now, $\lim _{x \rightarrow 0} f(x)$
$=\lim _{x \rightarrow 0} \frac{1-\cos 4 x}{8 x^2}$
$=\lim _{x \rightarrow 0} \frac{2 \sin ^2 2 x}{8 x^2}$
$=\lim _{x \rightarrow 0} \frac{\sin ^2 2 x}{4 x^2}$
$=\lim _{x \rightarrow 0}\left(\frac{\sin 2 x}{2 x}\right)^2=1$
Also, $f(0)=k$
Hence $k=1$

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