Let f : R R be a function given by f(x)= ll 1- 2 x x^2 & , x<0 & … - Vidyadip

MCQ

Let $f : R \rightarrow R$ be a function given by
$f(x)=\left\{\begin{array}{ll}\frac{1-\cos 2 x}{x^2} & , x<0 \\ \alpha & , x=0, \text { where } \alpha, \beta \in R . \text { If } \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x>0\end{array}\right.$
$f$ is continuous at $x = 0$, then $a^2 + B^2$is equal to:

A
48
B
12
C
3
D
6

✓

Answer

$f\left(0^{-}\right)=\lim _{x \rightarrow 0^{-}} \frac{2 \sin ^2 x}{x^2}=2=\alpha$
$f\left(0^{+}\right)=\lim _{x \rightarrow 0^{+}} \beta \times \sqrt{2} \frac{\sin \frac{x}{2}}{2 \frac{x}{2}}=\frac{\beta}{\sqrt{2}}=2$
$\Rightarrow \beta=2 \sqrt{2}$
$\alpha^2+\beta^2=4+8=12$

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