If f(x)= array|c|c| a|x|+x^2-2( |x|)( |x|)x & ,x 0 b & ,x=0 array is continuous at x=0, th…

Question

If $f(x)= \begin{array}{|c|c|} \frac{a|x|+{x}^{2}-2(\sin |x|)(\cos |x|)}{x} & ,x
e 0 \ b & ,x=0 \end{array}$

is continuous at $x=0$, then $a+b$ is equal to :

Accepted Answer

Refunctioning the given function.

$ $$\& f(x)={|c|c|}\frac{ax+{x}^{2}-\sin 2x}{x} \& ,x>0  \  \frac{-ax+{x}^{2}+\sin 2x}{x} \& ,xFor continuity:$$

$$$\begin{gathered} LHL=RHL\Rightarrow a-2=-a+2\Rightarrow a=2 \ b=f(0)=RHL=2-2=0 \ a+b=2+0=2 \end{gathered}$$

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