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Limits in Euclidean Space

  1. Feb 20, 2007 #1
    When dealing with real valued functions (one output for now) of more than one real variable, can the usual rules from R --> R be generalised in the natural way? Specifically the sum, product, quotient and composite rules. Any pathological cases?

    Also I was also wondering if there are any tools analogous to L'Hopital lying around anywhere.
  2. jcsd
  3. Feb 20, 2007 #2


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    Looks to me like you are talking about Rn--> R: a real valued function of several values. Yes, the standard rules for derivatives apply to partial derivatives and the gradient vector.
  4. Feb 21, 2007 #3
    Yes I know that, I was more concerned about actual limits at a pedantic epsilon-delta level. Eg. for the composite rule if f(x,y) = h(g(x,y)), and we know that

    lim g(x,y) = L

    then can we always say that

    lim f(x,y) = lim h(z)
    (x,y)-->(a,b) z-->L
  5. Feb 21, 2007 #4

    matt grime

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    Derivatives *are* pedantic epsilon and delta arguments. And they all pass through without alteration in the R^m case: just replace | | with || ||, the Euclidean distance in R^m.

    It is things like the mean value theorem that fail.
  6. Feb 21, 2007 #5


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    Hmm, looking back at your original post I don't know where I got the idea that you were talking about derivatives!

    Yes, the "composition" law applies to limits of functions of several variables.
  7. Feb 21, 2007 #6
    Thanks guys
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