Question

The values of $p$ and $q$ for which the function $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x}& ,x<0\\ q& ,x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}}& ,x>0\end{array}$
is continuous for all $x$ in $R$, are

• A. $p=\frac{1}{2},q=-\frac{3}{2}$
• B. $p=\frac{5}{2},q=\frac{1}{2}$
• C. $p=-\frac{3}{2},q=\frac{1}{2}$
• D. $p=\frac{1}{2},q=\frac{3}{2}$

Answer 1

The correct option is C $p=-\frac{3}{2},q=\frac{1}{2}$

$f\left(x\right)=\{\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x}& ,x<0\\ q& ,x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}}& ,x>0\end{array}$
$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0^{+}}{lim}\frac{\sin (p+1)x+\sin x}{x}=p+2$
$\underset{x\to 0^{-}}{lim}f\left(x\right)=\frac{1}{2}=p+2=q=\frac{1}{2}$
$p+2=\frac{1}{2},q=\frac{1}{2}$
$\therefore p=-\frac{3}{2},q=\frac{1}{2}$
Hence, option 'C' is correct.