let $\lim_{x \to a} f(x) =l \neq 0 $ and $\lim_{x \to a} g(x) =\text{Does not exist}$

Question

let $\lim_{x \to a} f(x) =l \neq 0 $ and $\lim_{x \to a} g(x) =\text{Does not exist}$ then prove that :

$$\lim_{x \to a} g(x)f(x)=\text{Does not exist}$$

Answer 1

By contradiction, suppose that there exists $\lim\limits_{x\to a} g(x) f(x)$. Because $\lim\limits_{x\to a} f(x)=l\neq 0$, then there exists $\lim\limits_{x\to a} \dfrac{g(x) f(x)}{f(x)}=\lim\limits_{x\to a} g(x)$, clearly, a contradiction.

Answer 2

It is well known that

If $lim_{x\rightarrow a}u(x)=\ell $ and $lim_{x\rightarrow a}v(x)=\ell'\neq 0$, than $$ lim_{x\rightarrow a}\frac{u}{v}(x)=\frac{\ell}{\ell'}$$

So, here, with $v=f$ and $u=gf$, if you assume that "$lim_{x\rightarrow a} g(x) f(x)$ does exist", it will imply that "$lim_{x\rightarrow a} g(x)$ does exist"...

So $lim_{x\rightarrow a} g(x) f(x)$ does not exist.