If f(x)=cases(p+1)x+ xx, & x0cases is continuous at x=0, then …

Question

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

Accepted Answer

LHL $=(p+1)+1=p+2$.

RHL $=\displaystyle\lim_{h	o0}\frac{h+h^2-\sqrt h}{h^{3/2}}=\lim_{h	o0}\frac{\sqrt h(\sqrt h+1)-1}{h}\cdot$ ... simplifying, $=\displaystyle\lim_{h	o0}\frac{\sqrt{h}+1-1}{\sqrt h+1+1}\cdot\frac{1}{1}=\frac12$.

Continuity: $p+2=q=\dfrac12\Rightarrow p=-\dfrac32,\ q=\dfrac12$.

Solutions · 1