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Continuity and Differentiability — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSContinuity and Differentiability1 Mark
Question
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
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Answer

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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