If $f(x) = \begin{cases} \frac{x}{e^{1/x} + 1}, & x \ne 0 \\ 0, & x = 0 \end{cases}$,then

Let $f$ be a continuous,periodic even function defined on $\mathbb{R}$ such that $f(0) = 1$,$f(2) = -1$ and the period of $f$ is $4$. The minimum number of roots of the equation $f(x) = 0$ in the interval $[-10, 10]$ will be:

If the function $f(x)=\begin{cases} \frac{\tan a(x-1)}{x-1}, & \text{if } 0 < x < 1 \\ \frac{x^3-125}{x^2-25}, & \text{if } 1 \leq x \leq 4 \\ \frac{b^x-1}{x}, & \text{if } x > 4 \end{cases}$ is continuous in its domain, then $6a + 9b^4 = $

If $f(x) = \begin{cases} 3 + x; & x \geqslant 0 \\ 2 - 3x; & x < 0 \end{cases}$,then $\lim_{x \to 0} f(f(x))$ is equal to -

The number of discontinuities of the greatest integer function $f(x) = [x]$ for $x \in \left(-\frac{7}{2}, 100\right)$ is:

Let $R$ be the set of all real numbers and $\alpha \in R$ be positive. Define a function $f: R \rightarrow R$ by $f(0)=0$ and $f(x)=|x|^\alpha \sum \limits_{n=0}^{\infty}\left(1+x^2\right)^{-n}$,for $x \neq 0$. Then the set of real numbers $\alpha$ for which $f$ is continuous at $x = 0$ has