[RESOLVED] Limit

12052016, 09:38 PM
(This post was last modified: 12052016 09:40 PM by Han.)
Post: #12




RE: Limit
Be careful about assuming that a complex limit is equivalent to a double limit over real variables. The following limit does not exist.
\[ \lim_{(x,y)\to (0,0)} \left( \frac{x^2y^2}{x^2+y^2} \right)^2 \] Reason: \[ \lim_{(x,0)\to (0,0)} \left( \frac{x^2y^2}{x^2+y^2} \right)^2 = \lim_{(x,0)\to (0,0)} \left( \frac{x^20}{x^2+0} \right)^2 = 1\] \[ \lim_{(x,x)\to (0,0)} \left( \frac{x^2y^2}{x^2+y^2} \right)^2 = \lim_{(x,x)\to (0,0)} \left( \frac{x^2x^2}{x^2+x^2} \right)^2 = 0\] (12052016 12:17 PM)DrD Wrote: Thinking about this further, is it fair to say that as long as "regardless of the order" of the variables always converges to the same limit value, the limit exists, but if reordering the variables does NOT converge to the same limit, that a limit does not exist? The example above shows that "swapping order" is not sufficient as the squaring of the fraction produces a limit of 1 even after interchanging x and y. Graph 3D  QPI  SolveSys 

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Messages In This Thread 
[RESOLVED] Limit  jrozsas  12012016, 10:43 AM
RE: Limit  Nigel (UK)  12022016, 10:49 AM
RE: Limit  Tim Wessman  12052016, 07:21 AM
RE: Limit  Han  12052016 09:38 PM

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