Multivariable Limits

  1. I was presented with the two following questions:

    [tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} \sin\frac{xy}{xy}[/tex]


    [tex]\lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} \sin(\frac{xyz}{xyz})[/tex]

    I figured I would do a simple substitution: let t = xy for the first one, and the limit becomes as t ->0, sin t /t would approach 1. The answer is right for the first one.

    Why doesn't the same technique work for the second one? (The answer for the second one is 0).
    Last edited: Sep 21, 2006
  2. jcsd
  3. 0rthodontist

    0rthodontist 1,249
    Science Advisor

    In the second one are you also meant to have the limit be as z goes to 0?

    Whenever x, y, and z are all nonzero, then xyz/(xyz) = 1. So sin(xyz/(xyz)) = sin 1. This does not depend on the path that you approach (0, 0, 0) by, so the limit is sin 1.
  4. HallsofIvy

    HallsofIvy 41,267
    Staff Emeritus
    Science Advisor

    No, the way you've written it, the lilmit for the second one is not 0. It is, as Orthodontist said, sin 1, exactly like the first.
  5. Sorry. I'm very bad with using latex.

    I actually meant [sin (xy )]/(xy)

    and likewise for the second one. For the second one, there's also a z->0 which I omitted. Sorry again.
  6. 0rthodontist

    0rthodontist 1,249
    Science Advisor

    In that case your reasoning is correct and the limit is 1, not 0. Why did you think it is 0? (maybe you're using your calculator to check and you're using degrees instead of radians?)
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