WBJEE · Maths · Continuity and Differentiability
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 \(\mathrm{a}=-\frac{5}{2}, \mathrm{~b}=-\frac{3}{2}, \mathrm{c}=\frac{3}{2}\)
- D \(a=-2, b \in \mathbb{R}-\{0\}, c=0\)
Answer & Solution
Correct Answer
(D) \(a=-2, b \in \mathbb{R}-\{0\}, c=0\)
Step-by-step Solution
Detailed explanation
\(\quad f\left(0^{+}\right)=\underset{x \rightarrow 0^{+}}{L t} \frac{(1+b x)^{1 / 2}-1}{b}=0\) \(f(0)=c\) \(f\left(0^{-}\right)=a+2\) \(\Rightarrow c=0\) and \(a=-2, b \in R-\{0\}\)
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