Let f(x)=casesa|x|+x^2-2|x||x|x, & x0 b, & x=0cases be continuous at x=0.

Question

Let $f(x)=\begin{cases}\dfrac{a|x|+x^2-2\sin|x|\cos|x|}{x}, & x
eq0\ b, & x=0\end{cases}$ be continuous at $x=0$. Then $a+b$ is equal to:

Accepted Answer

For $x>0$: $f(x)=a+x-\dfrac{\sin2x}{x}$, so
$$\text{RHL}=\lim_{x\to0^+}\left(a+x-2\cdot\frac{\sin2x}{2x}\right)=a-2$$

For $x<0$: $|x|=-x$ and $\sin(2|x|)=-\sin2x$, giving $f(x)=-a+x+\dfrac{\sin2x}{x}$, so
$$\text{LHL}=\lim_{x\to0^-}\left(-a+x+2\cdot\frac{\sin2x}{2x}\right)=-a+2$$

Continuity requires LHL $=$ RHL: $-a+2=a-2\implies a=2$. Then $b=f(0)=\text{RHL}=2-2=0$.

$a+b=2+0=2$

Solutions · 1