Answer
$\lim\limits_{x \to 0}f(x)=4$
Work Step by Step
(a) We start by graphing the function $f(x)=\frac{\tan (4x)}{x}$ and looking at the part of the graph around the $y$-intercept.
We can see clearly from the graph that $f(0)$ approaches $4$ when $x\rightarrow 0$ from both sides.
(b) The function is undefined for $x = 0$ because $\frac{\tan(0)}{0}$ is undefined.
We can evaluate using values very close to $0$ to find the left and right hand limits.
$\lim\limits_{x \to 0^{-}}f(x)\approx{f(-0.00001)=\frac{\tan(4(-0.00001))}{(-0.00001)}}=4$
$\lim\limits_{x \to 0^{+}}f(x)\approx{f(0.00001)=\frac{\tan(4(0.00001)}{0.00001}}=4$
So, $\lim\limits_{x \to 0}f(x)=4$.
