[RESOLVED] Limit

[Han]

Be careful about assuming that a complex limit is equivalent to a double limit over real variables. The following limit does not exist.

\[ \lim_{(x,y)\to (0,0)} \left( \frac{x^2-y^2}{x^2+y^2} \right)^2 \]

Reason:

\[ \lim_{(x,0)\to (0,0)} \left( \frac{x^2-y^2}{x^2+y^2} \right)^2
= \lim_{(x,0)\to (0,0)} \left( \frac{x^2-0}{x^2+0} \right)^2
= 1\]

\[ \lim_{(x,x)\to (0,0)} \left( \frac{x^2-y^2}{x^2+y^2} \right)^2
= \lim_{(x,x)\to (0,0)} \left( \frac{x^2-x^2}{x^2+x^2} \right)^2
= 0\]

(12-05-2016 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.