Answer
a) $\lim\limits_{x \to \infty}x\sin(\frac{1}{x})=1$
b) $\lim\limits_{x \to 0}x\sin(\frac{1}{x})=0$
c) See image
Work Step by Step
a) $\lim\limits_{x \to \infty}x\sin(\frac{1}{x})$
Let's take a variable $a=\frac{1}{x}$
$\lim\limits_{x \to \infty}x\sin(\frac{1}{x})=\lim\limits_{a \to 0}\frac{1}{a}\sin(a)=\lim\limits_{a \to 0}\frac{\sin(a)}{a}$
We know that the limit above is 1.
b) $\lim\limits_{x \to 0}x\sin(\frac{1}{x})=\lim\limits_{x \to 0}x\times\lim\limits_{x \to 0}\sin(\frac{1}{x})=0\times\lim\limits_{x \to 0}\sin(\frac{1}{x})=0$
c) The function approaches 1 as x tends to infinity and approaches 0 as x tends to 0. See image
