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Multivariable Limit (Definition of Derivative)

  1. Oct 20, 2008 #1
    1. The problem statement, all variables and given/known data
    I need to show that |sin(e^xy)-sin(1)|/(x^2+y^2)^1/2 -> 0 as (x,y) -> (0,0)

    2. Relevant equations
    Triangle Inequality?

    3. The attempt at a solution
    I know that this is true, since e^xy -> 1 as (x,y) -> (0,0) much, much faster than (x^2+y^2)^1/2 -> 0 as (x,y) -> (0,0). I don't know how to give this limit an upper bound to prove it though. Otherwise, I guess I could use an epsilon-delta proof, but I think that might be a little much?
  2. jcsd
  3. Oct 20, 2008 #2


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    Homework Helper

    Recall that [tex]f^{\prime}(x)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}[/tex].
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