Answer
$\lim\limits_{x\to 0}\dfrac{\tan^2{x}}{x}=0.$
Work Step by Step
Using Theorem $1.9:$
$\lim\limits_{x\to 0}\dfrac{\tan^2{x}}{x}=\lim\limits_{x\to 0}\dfrac{\tan{x}}{x}\times\lim\limits_{x\to 0}\tan{x}=\lim\limits_{x\to 0}\dfrac{\dfrac{\sin{x}}{\cos{x}}}{x}\times\lim\limits_{x\to 0}\tan{x}$
$=\lim\limits_{x\to 0}\dfrac{\sin{x}}{x}\times\lim\limits_{x\to 0}\dfrac{1}{\cos{x}}\times\lim\limits_{x\to 0}\tan{x}=(1)(1)(0)=0.$
