Question

If $f(x) = \begin{cases} \frac{\sin (p +1) + \sin x}{x} &, \quad x < 0 \\ q &, \quad x = 0 \\ \frac{\sqrt{x+x^2} - \sqrt{x}}{x^{3/2}} &, \quad x > 0 \end{cases}$
is continuous at $x = 0$, then the ordered pair $(p,q)$ is equal to :

• A. $\left(\frac{5}{2} , \frac{1}{2}\right)$
• B. $\left( - \frac{3}{2} , - \frac{1}{2}\right)$
• C. $\left( - \frac{1}{2} , \frac{3}{2}\right)$
• D. $\left( - \frac{3}{2} , \frac{1}{2}\right)$

Answer 1

Answer: (D)

$RHL = \lim_{x\to0^{+}} \frac{\sqrt{x+x^{2}} -\sqrt{x}}{x^{\frac{3}{2}}} =\lim_{x\to0^{+} } \frac{\sqrt{1+x}-1}{x} = \frac{1}{2} $
$ LHL = \lim_{x\to0} \frac{\sin\left(p+1\right)x+\sin x}{x} =\left(p+1\right)+1 = p+2 $ for continuity $ LHL = RHL = f\left(0\right) $
$ \Rightarrow \left(p,q\right) = \left(\frac{-3}{2} , \frac{1}{2} \right)$