If $f(x) = \frac{e^{x^2} - \cos x}{x^2}$ for $x \neq 0$ is continuous at $x = 0$,then the value of $f(0)$ is

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

If the function $f(x) = \begin{cases} \frac{1}{x} \log_{e}\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right), & x < 0 \\ k, & x = 0 \\ \frac{\cos^{2} x - \sin^{2} x - 1}{\sqrt{x^{2}+1}-1}, & x > 0 \end{cases}$ is continuous at $x = 0$,then $\frac{1}{a} + \frac{1}{b} + \frac{4}{k}$ is equal to:

$f(x) = \begin{cases} \frac{x-4}{|x-4|} + a, & x < 4 \\ a+b, & x=4 \\ \frac{x-4}{|x-4|} + b, & x > 4 \end{cases}$
If $f(x)$ given above is continuous at $x=4$,then find the values of '$a$' and '$b$'.

Consider the function $f(x) = \begin{cases} \frac{x}{[x]} & \text{if } 1 \leqslant x < 2 \\ 1 & \text{if } x = 2 \\ \sqrt{6-x} & \text{if } 2 < x \leqslant 3 \end{cases}$ where $[x]$ denotes the greatest integer function. At $x = 2$,the function:

Let $f: R \rightarrow R$ be defined as:
$f(x) = \begin{cases} \frac{\lambda|x^{2}-5x+6|}{\mu(5x-x^{2}-6)}, & x < 2 \\ \mu, & x = 2 \\ e^{\frac{\tan(x-2)}{x-[x]}}, & x > 2 \end{cases}$
Where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x = 2$,then $\lambda + \mu$ is equal to: