Answer
This is true.
Work Step by Step
If $$\lim_{(x,y)\to(0,0)}f(x,y)=0,$$
then by deffinition we have that fo every $\varepsilon>0$ there is $\delta>0$ such that when $\sqrt{x^2+y^2}0$ there is $\delta>0$ such that when $\sqrt{x^2+0^2}0$ there is $\delta >0$ such that when $|x|$
