Show that there is real $r$ such that for every real $x$ we have $l\cdot f(x)>0$

Question

Given $$\lim_{x\to x_0}f(x)=l$$ and $l\neq 0$ show that :

there is a real $r>0$ for every $x\in (x_0-r,x_0+r)-\{x_0\}$

such that : $f(x)\cdot l>0$

Should I use the delta epsilon definition of limit to show it ?

So we have to show that the function and the limit have the same sign .