We have two functions $f(x)$ and $g(x)$, such that $\lim_{ x\to 0}f(x)=0$ but $\lim_{x\to 0} g(x)$ does not exist.
Would that mean that $\lim_{x\to 0}f(x)g(x)= 0$, assuming we don't divide by zero anywhere?
Also, is this true in the multivariable case, where $f$ is a function of $x$ and $g$ is a function of $y$? In that case, is $\lim_{(x,y)\to(0,0)}=f(x)g(y)=0$?
If $g(x)$ is bounded in a neighbourhood of $0$, then this is true. If $g(x)$ is not bounded in a neighbourhood of $0$, then this in not true in general. Take for instance $f(x)=x$ and $g(x)=\ln(x).$ Then $\lim_{x\to 0} f(x)g(x)=0$.
On the other hand we can take $f(x)=\dfrac{1}{\ln(x)}$ and $g(x)=\big(\ln(x)\big)^2$ and we have $\lim_{x\to 0}f(x)g(x)=-\infty$.
We can do the same in the two-variable case. $\lim_{(x,y)\to (0,0)} f(x)g(y)=0$ if $g(y)$ is bounded in some neighbourhood of $0$. If it is not then this is not true in general, because we can always approach $(0,0)$ along the path $x=y$.
