# Is it true that $\lim_{x\to 0}\lim_{y \to 0}f(x,y) = \lim_{x\to 0}f(x,x)$?

Is it true that $$\lim_{x\to 0}\lim_{y \to 0}f(x,y) = \lim_{x\to 0}f(x,x)?$$

I have no idea how to disprove or prove it, but it seems intuitively like it must be true.

• I don't think so. You are talkin about two different variables. – The Dead Legend Apr 11 '17 at 5:41
• But do you mean the limit $(x,y)\to (0,0)$? – Vincenzo Zaccaro Apr 11 '17 at 9:44

No. Consider this counter example: $$\lim_{x\to 0}\lim_{y\to 0} \frac{y^2}{xy}=\lim_{x\to 0}0=0$$ while $$\lim_{x\to 0}\frac{x^2}{x^2}=1.$$
In general it is not true because you talking about two different variables because if you consider the function $$f(x,y) = \frac{xy-xy^2}{x-y}$$ then you see that $$\lim{x\to 0}\lim{y\to 0}f(x,y)$$ gives defined value if you want to replace x with y then function is not defined So it may be true in few cases but not in general