If $f(x) = \frac{x^2-10x+25}{x^2-7x+10}$ and $f$ is continuous at $x=5$,then $f(5)$ is equal to

$f(x) = \begin{cases} \frac{e^{\alpha x} - e^{x} - x}{x^{2}}, & x \neq 0 \\ \frac{3}{2}, & x = 0 \end{cases}$ Find the value of $\alpha$ for which the function $f$ is continuous.

Let $S_n = 1 + 3x + 9x^2 + 27x^3 + \ldots$ ($n$ terms) and $-\frac{1}{3} < x < \frac{1}{3}$. If $\lim_{n \rightarrow \infty} S_n = f(x)$,then $f(x)$ is discontinuous at the point $x =$

Number of points of discontinuity of the function $f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\})$ for $x \in [0, 4]$ is (where $[.]$ and $\{.\}$ denote the greatest integer and fractional part functions,respectively).

If $f(x) = \begin{cases} \frac{1-\sin^3 x}{3 \cos^2 x}, & x < \frac{\pi}{2} \\ \alpha, & x = \frac{\pi}{2} \\ \frac{\beta(1-\sin x)}{(\pi-2 x)^2}, & x > \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$,then $\alpha \beta =$

If $[x]$ is the greatest integer function and $f(x) = \begin{cases} 2[x] - \frac{x}{|x|}, & x \neq 0 \\ 1, & x = 0 \end{cases}$ is a real-valued function,then $f$ is