Q50 / 155 · WBJEE 2022 · Mathematics

The values of $a, b, c$ for which the function $f(x)=\left\{\begin{array}{l}\frac{\sin (a+1) x+\sin x}{x}, x < 0 \\ c, x=0 \\ \frac{\left(x+b x^2\right)^{1 / 2}-\sqrt{x}}{b x^{1 / 2}}, x>0\end{array}\right.$ is continuous at $x=0$, are

  • A
    $a=\frac{3}{2}, b=-\frac{3}{2}, c=\frac{1}{2}$
  • B
    $\mathrm{a}=-\frac{3}{2}, \mathrm{c}=\frac{3}{2}, \mathrm{~b}$ is arbitrary non-zero real number
  • C
    $\mathrm{a}=-\frac{5}{2}, \mathrm{~b}=-\frac{3}{2}, \mathrm{c}=\frac{3}{2}$
  • Correct answer
    D
    $a=-2, b \in \mathbb{R}-\{0\}, c=0$

Solution

$\quad f\left(0^{+}\right)=\underset{x \rightarrow 0^{+}}{L t} \frac{(1+b x)^{1 / 2}-1}{b}=0$ $f(0)=c$ $f\left(0^{-}\right)=a+2$ $\Rightarrow c=0$ and $a=-2, b \in R-\{0\}$

Hence, the correct answer is option: 'D'.

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