proof writing – Question about limits at infinity.

if for example I have fucntions $f$ and $g$, we’ve prove in class that if
$$lim _{xto a}left(fleft(xright)-gleft(xright)right)=L$$


$$lim _{xto a}left(fleft(xright)cdot gleft(xright)right)=L$$

then both limits of $f$ and $g$ exist, or both don’t exist.

but what about

$$lim _{xto ∞/-∞}left(fleft(xright)-gleft(xright)right)=L$$


$$lim _{xto ∞/-∞}left(fleft(xright)cdot gleft(xright)right)=L$$

Does it still hold that both of f,g must exist or both must not exist?
If so, how do I prove it?