If $\lim_{x \to a}f(x)$ and $\lim_{x \to a} (f(x)g(x))$ exist, must $\lim_{x \to b} g(x)$ exist?
Let $f,g$ be two functions defined in some open interval $I$ containing a point $a$. Let us suppose that $a=b$.
Let $\epsilon>0$. We know that $\lim_{x \to a}f(x)$ and $\lim_{x \to a}(f(x)g(x))$ exist.
Therefore, let $\lim_{x \to a}f(x)=l$ and $\lim_{x \to a}(f(x)g(x))=l'$.
Since $\lim_{x \to a}f(x)=l$ and $\lim_{x \to a}(f(x)g(x))=l'$, there exists $\eta_1>0$ and $\eta_2>0$ such that:
$\forall x \in I $ \begin{array}{lcl} \mid x-a \mid \leq \eta_1 \Rightarrow \mid f(x)-l \mid \leq \\ \mid x-a \mid \leq \eta_2 \Rightarrow \mid f(x)g(x)-l' \mid \leq \epsilon\\ \end{array}
I didn't find the first line has to be smaller that and how to I reveal the $g(x)$?
Thank you in advance.