Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor

    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

    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook