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Solving limit problems for two variables

  1. Sep 22, 2012 #1
    I just encountered a limit-problem [1] which is supposed to lack a limit:

    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.
     
  2. jcsd
  3. Sep 22, 2012 #2
    [itex] (x+y)²=x²+y²+2xy [/itex]. we know that [itex] x²+y²= ∞ [/itex]
    However, this means that there are many possibilities: x = ± ∞ or y=±∞. there are the following possibilities: x=+∞ and y=+∞ ; x=+∞ and y=-∞ ; x= + ∞ and y is a finite number greater than 0 ; x = + ∞ and y is a finite number smaller than 0 ; x=-∞ a and y= -∞ ; x= - ∞ and y= + ∞ ; x= - ∞ and y is some finite number greater than 0; x = - ∞ and y is some finite number smaller than 0. and of courses all the cases where x is some finite number (negative or positive).
    so not only could [itex]x²+y²+2xy[/itex] be + ∞ or - ∞, it might also be undefined. (for instance when 2xy=-∞). or the whole thing may be undefined if x=0 and y=infinity.
    hence I would presume that to know the limit is impossible.
     
    Last edited: Sep 22, 2012
  4. Sep 22, 2012 #3

    LCKurtz

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    Actually it relies more on experience.

    What happens if ##x\rightarrow \infty## along the line ##y=0##? In order to get rid of that negative exponential, you might try the limit along the line ##y=-x##. What happens there?
    Usually, if the limit doesn't exist, it isn't too difficult to find a couple of paths where the answers differ, which is all you need to show the limit doesn't exist.
    Sometimes you can find an appropriate inequality or use a squeeze theorem or use something like your suggestion of polar coordinates. There is no one general method. What is normally done while learning the topic is to work some "easy" ones. Then you develop the limit theorems and continuity theorems and use them.
     
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