- #1

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x*y*e^(-(x+y)^2)

as x^2+y^2 approaches infinity and e is the base of the natural logarithm.

My approach was to express x and y in terms of polar coordinates, where x = r*cos(t) and y= r*sin(t). I got the following expression:

[(r^2)*cos(t)*sin(t)]/[e^(r^2)(1+cos(t)*sin(t))]

I don't see how it would be possible for me to deduce out of this expression that it lacks a limit. I do know that in general, one can refute the existence of the limit of an expression if one gets different limits when studying different curves/lines that intersect through the point (a,b) to which the point (x,y) approaches.

The problem with this method is that it largely relies on LUCK AND INTUITION. At least, that's the impression I get through the textbooks and lectures that I've had so far. I have not been introduced to a systematic approach. I would like someone to:

-Explain one way to show how this particular limit problem [1] lacks a limit.

-Explain if there are any systematic (or somewhat systematic) ways to study limits of two variables.

-Eventually explain more general methods that would apply for studies of limits of an arbitrary number of variables.