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Multidimensional limits

  1. Mar 5, 2006 #1
    I'm trying to understand how to interpret multidemensional limits. For example, suppose you have the following:

    [tex]\lim\limits_{x \to \infty}\lim\limits_{y \to \infty} x\frac{1}{y}[/tex]

    Would this be infinity, 0, or 1?

    This is really a more general version of the question I'm working with regarding the behavior of a function that has the property [tex]f_a(b) \to\limits_{a \to \infty} 1[/tex] and [tex]f_a(b) \to\limits_{b \to \infty} \infty[/tex] in the context that both a and b are going to infinity.

    Thanks in advance for any help!
     
  2. jcsd
  3. Mar 6, 2006 #2

    Galileo

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    Well,
    [tex]\lim \limits_{x \to \infty}\left(\lim\limits_{y \to \infty} \frac{x}{y}\right)=\lim \limits_{x \to \infty} 0=0[/tex]
    ,but
    [tex]\lim\limits_{y \to \infty}\left(\lim\limits_{x \to \infty} \frac{x}{y}\right
    )[/tex]
    doesn't exist, because [itex]\lim\limits_{x \to \infty} \frac{x}{y}[/itex] doesn't exist.

    So that shows that, in general, you can't just interchange limits. They don't commute.
     
  4. Mar 6, 2006 #3

    HallsofIvy

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    None of the above! The limit simply doesn't exist.

    That means there are points arbitrarily far from the origin such that f is close to 1 and also points such that f is arbitrarily large. There is no one number (not even [itex]\infty[/itex]) that the function gets close to. If there are two different limits by approaching a given point (even "the point at infinity" in two different ways, then the limit itself does not exist.
     
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