If $f(x) = \begin{cases} \frac{8^x - 4^x - 2^x + 1}{x^2} & , \text{if } x > 0 \\ e^x \sin x + kx + \lambda \log 4 & , \text{if } x \le 0 \end{cases}$ is continuous at $x = 0$,then the value of $500 e^\lambda$ 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

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x) = \begin{cases} b^2 \sin \left(\frac{\pi}{2} \left[\frac{\pi}{2}(\cos x + \sin x) \cos x\right]\right), & x < 0 \\ \frac{\sin x - \frac{1}{2} \sin 2x}{x^3}, & x > 0 \\ a, & x = 0 \end{cases}$ is continuous at $x = 0$,then $a^2 + b^2$ is equal to

If $f(x) = \begin{cases} \frac{x^3 + x^2 - 16x + 20}{(x-2)^2}, x \neq 2 \\ k, x = 2 \end{cases}$ is continuous at $x = 2$,then $k = \rule{1cm}{0.15mm}$

Discuss the continuity of the $sine$ function.

If $f(x) = \begin{cases} \frac{5}{2} - x, & x < 2 \\ 1, & x = 2 \\ x - \frac{3}{2}, & x > 2 \end{cases}$,then: