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Continuity — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsContinuity2 Marks
MCQ
If $f(x)$ is continuous in $[-2,2]$, where
$f(x)=\left\{\begin{array}{ll}\frac{\sin a x}{x}-2, & \text { for }-2 \leq x < 0 \\2 x+1, & \text { for } 0 \leq x \leq 1 \\2 b \sqrt{x^2+3}-1, & \text { for } 1 < x \leq 2\end{array}\right.$
then the value of $(a+b)$ is
  • A
    2
  • ✓
    4
  • C
    6
  • D
    8
✓

Answer

Correct option: B.
4
(B)
Since $f (x)$ is continuous in $[-2,2]$.
∴ $\quad$ it is continuous at $x=0$ and $x=1$.
$\therefore \quad \lim _{x \rightarrow 0^{-}} f (x)=\lim _{x \rightarrow 0^{+}} f (x)$
$\Rightarrow \lim _{x \rightarrow 0^{-}}\left(\frac{\sin a x}{x}-2\right)=\lim _{x \rightarrow 0^{+}}(2 x+1)$
$\Rightarrow a-2=0+1 \Rightarrow a=3$
Also, $\lim _{x \rightarrow 1^{-}} f (x)=\lim _{x \rightarrow 1^{+}} f (x)$
$\Rightarrow \lim _{x \rightarrow 1^{-}}(2 x+1)=\lim _{x \rightarrow 1^{+}}\left(2 b \sqrt{x^2+3}-1\right)$
$\Rightarrow 2(1)+1=2 b \sqrt{1+3}$
$\Rightarrow 3=4 b-1$
$\Rightarrow b =1$
$\therefore \quad a+b=3+1=4$

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