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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.

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  • $\begingroup$ I don't think so. You are talkin about two different variables. $\endgroup$ – The Dead Legend Apr 11 '17 at 5:41
  • $\begingroup$ But do you mean the limit $(x,y)\to (0,0)$? $\endgroup$ – Vincenzo Zaccaro Apr 11 '17 at 9:44
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No. Consider this counter example: $$\lim_{x\to 0}\lim_{y\to 0} \frac{y^2}{xy}=\lim_{x\to 0}0=0$$ while $$\lim_{x\to 0}\frac{x^2}{x^2}=1.$$

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  • $\begingroup$ Ah so it is not true intuitively because in the first case tbeelimit is first taken w.r.t. 1 variable, and then w.r.t the second when the value of the first limit is already inserted into the formula, whereas in the second case the limits are basiclly taken "at the same time" $\endgroup$ – user56834 Apr 11 '17 at 10:20
  • $\begingroup$ Right, in general limits don't commute. $\endgroup$ – Couchy Apr 11 '17 at 16:38
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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

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