Mathematics - Limits Question with Solution | TestHub

Let the function $f:[1,\infty )\to \mathbb{R}$ be defined by $f(t)=\{\begin{array}{cc}(-1{)}^{n+1}2,& \text{ if }t=2n-1,n\in \mathbb{N},\\ \frac{(2n+1-t)}{2}f(2n-1)+\frac{(t-(2n-1))}{2}f(2n+1),& \text{ if }2n-1<t<2n+1,n\in \mathbb{N}.\end{array}$ Define $g(x)={\int }_1^{x}f(t)dt,x\in (1,\infty )$. Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta ={\lim }_{x\to 1^{+}}\frac{g(x)}{x-1}$. Then the value of $\alpha +\beta$ is equal to ________.