If $f(x) = \begin{cases} x \left(1 + \frac{1}{2} \sin (\log x^2) \right), & x \neq 0 \\ 0, & x = 0 \end{cases}$,then find the value of $\lim_{x \rightarrow 0} \frac{f(x) - f(0)}{x}$.

If the function $g(x) = \begin{cases} k\sqrt{x+1}, & 0 \le x \le 3 \\ mx + 2, & 3 < x \le 5 \end{cases}$ is differentiable,then the value of $k+m$ is:

Let $f: R \rightarrow R$ and $g: R \rightarrow R$ be respectively given by $f(x)=|x|+1$ and $g(x)=x^2+1$. Define $h: R \rightarrow R$ by $h(x)=\begin{cases} \max \{f(x), g(x)\} & \text{if } x \leq 0 \\ \min \{f(x), g(x)\} & \text{if } x > 0 \end{cases}$. The number of points at which $h(x)$ is not differentiable is

Let $f(x) = \begin{cases} -1, & -2 \le x < 0 \\ x^2 - 1, & 0 \le x \le 2 \end{cases}$ and $g(x) = |f(x)| + f(|x|)$. Then,in the interval $(-2, 2)$,$g$ is

If $x + |y| = 2y,$ then $y$ as a function of $x,$ at $x = 0$ is

$f(x) = |\log_e |x||$ is differentiable at