The values of p and q for which the function f(x)=cases(p+1)x+ xx, & x

Question

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

Accepted Answer

LHL $=(p+1)+1=p+2$. RHL $=\displaystyle\lim_{x	o0^+}\dfrac{x+x^2-x}{x^{3/2}(\sqrt{x+x^2}+\sqrt x)}=\dfrac{1}{1+1}=\dfrac12$.

$f(0)=q$. Continuity: $p+2=q=\dfrac12\Rightarrow p=-\dfrac32,\ q=\dfrac12$.

Solutions · 1