JEE Advanced 2024 Mathematics question on Definite Integrals and Areas

Question

Let the function

f:[1,\infty) \to \mathbb{R}

be defined by

f(t) = \begin{cases} (-1)^{n+1}2, & \text{if } t = 2n-1,\ n \in \mathbb{N}, \\ \dfrac{(2n+1-t)}{2}\,f(2n-1) + \dfrac{(t-(2n-1))}{2}\,f(2n+1), & \text{if } 2n-1 < t < 2n+1,\ n \in \mathbb{N}. \end{cases}

Define

g(x) = \displaystyle\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 = \displaystyle\lim_{x \to 1^+} \dfrac{g(x)}{x-1}

. Then the value of

\alpha + \beta

is equal to _____.