Limits, Continuity and Differentiability

If \(f(x)=\frac{|x|}{x}\), for \(x \neq 0\)
\(=1\), for \(x=0\), then tre function is

The function \(f(x)=\frac{x+1}{9 x+x^3}\) is

$$\begin{aligned} & \text { If } f(x)=\left[\tan \left(\frac{\pi}{4}+x\right)\right]^{\frac{1}{x}}, \quad x \neq 0 \\ & =k \text {, } \qquad x=0 \text { is continuous }\\ & x=0 \end{aligned}$$ Then $k=$

If $f(x)$ is continuous at $x=3$, where
$$\begin{aligned} f(x) & =a x+1, & \text { for } x \leq 3 \\ & =b x+3 & , \text { for } x>3 \text { then } \end{aligned}$$

If the function $f(x)=\frac{\log (1+a x)-\log (1-b x)}{x}$ $x \neq 0$ is continuous at $x=0$ then, $f(0)=\ldots \ldots$

Which of the following function is not continuous at $x=0$ ?

If function
$$\begin{aligned} f(x) & =x-\frac{|x|}{x}, x<0 \\ & =x+\frac{|x|}{x}, x>0 \\ & =1, \quad x=0, \text { then } \end{aligned}$$

If $f(x)=[x]$, where $[x]$ is the greatest integer not greater than $x$, then $f^{\prime}\left(1^{+}\right)=$ ...........

If the function $f(x)=\frac{\left(e^{k x}-1\right) \tan k x}{4 x^2}, x \neq 0$
\(\qquad \qquad=16 \qquad x=0\)
is continuous at $x=0$, then $k=\ldots \ldots$