Answer
Please see below.
Work Step by Step
We prove the statement by contradiction.
Suppose that $\lim_{x \to c}f(x)=L$, where $L$ is a real number. So, by Theorem 1.2 (3) we have$$\lim_{x \to c} \left ( \vphantom{\frac{1}{f(x)}} f(x) \right )\left (\frac{1}{f(x)}\right )=(L)(0)=0 .$$But, this contradicts the fact that$$\left ( \vphantom{\frac{1}{f(x)}} f(x) \right )\left (\frac{1}{f(x)}\right )=1 \quad \Rightarrow \quad \lim_{x \to c}1=1 .$$Thus, $\lim_{x \to c}f(x)$ does not exist.
