Question

$f(x) = \left\{ \begin{matrix} \frac{a|x| + x^{2} - 2sin|x|cos|x|}{x} & ;x \neq 0 \\ b & ;x = 0 \end{matrix} \right.\ $

for continuity

$$\begin{matrix} & \ \lim_{x \rightarrow 0^{-}}\mspace{2mu} f(x) = \lim_{x \rightarrow 0^{+}}\mspace{2mu} f(x) = f(0) \\ & \ \lim_{x \rightarrow 0^{-}}\mspace{2mu}\frac{ah + h^{2} - 2(sinh)cosh}{- h} \\ & \ = \lim_{x \rightarrow 0^{+}}\mspace{2mu}\frac{ah + h^{2} - 2(sinh)cosh}{h} \\ & \text{~}\text{or}\text{~} - a + 2 = a - 2 = b \\ & 2a = 4 \\ & a = 2,b = 0 \\ & \ \therefore a + b = 2 \end{matrix}$$