MCQ
$\mathop {\lim }\limits_{x \to 0} \frac{{\sin ({x^{\frac{1}{3}}})\,\ln (1 + 3x)}}{{{{({{\tan }^{ - 1}}\sqrt x )}^2}\left( {{e^{5{x^{\frac{1}{3}}}}} - 1} \right)}} = $
- ✓$\frac{3}{5}$
- B$\frac{1}{5}$
- C$\frac{2}{5}$
- D$\frac{5}{3}$
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$f(t)=\left\{\begin{array}{cc}(-1)^{n+1} 2, & \text { if } t=2 n-1, n \in N , \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2 n-1 < t < 2 n+1, n \in N \end{array}\right.$
Define $g(x)=\int_1^x f(t) d t, x \in(1, \infty)$. Let $\alpha$ denote the number of solutions of the equation $g(x)=0$ in the interval $(1,8]$ and $\beta=\lim _{x \rightarrow 1+} \frac{g(x)}{x-1}$. Then the value of $\alpha+\beta$ is equal to. . . . . .