If $f(x) = \begin{cases} \frac{8^x-4^x-2^x+1}{x^2}, & \text{if } x > 0 \\ e^x \sin x + x + \lambda \log 4, & \text{if } x \leqslant 0 \end{cases}$ is continuous at $x = 0$,then the value of $1000 e^\lambda = $

Let $f(x) = \begin{cases} -2 \sin x, & \text{if } x \leq -\frac{\pi}{2} \\ A \sin x + B, & \text{if } -\frac{\pi}{2} < x < \frac{\pi}{2} \\ \cos x, & \text{if } x \geq \frac{\pi}{2} \end{cases}$. For what values of $A$ and $B$ is $f$ continuous?

If $f:(-7,7) \rightarrow R$ is defined by $f(x)=[x]$ for all $x \in (-7,7)$,then the number of discontinuities of $f$ is

Discuss the continuity of the function $f$ given by $f(x) = |x|$ at $x = 0$.

Let $f$ and $g$ be real-valued functions. If $\lim _{x \rightarrow 0} \frac{2 f(x)-g(x)}{[f(x)+7]^{2 / 3}}=\frac{7}{4}$, $\lim _{x \rightarrow 0} f(x)=1$ and $\lim _{x \rightarrow 0} g(x)=\alpha$, then $h(x)= \begin{cases} \sin (\alpha x), & 0 \leq x \leq \frac{\pi}{10} \\ \cos (2 \alpha x), & \frac{\pi}{10} < x \leq \frac{\pi}{5} \end{cases}$ is:

Examine the following function for continuity: $f(x) = x - 5$.