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Continuity of functions

  1. Nov 17, 2012 #1
    hi everyone, I've found this exercise on a text book and it doesn't resemble any exercise I've seen before. I just want to know how to proceed, you don't have to solve it for me :)

    1. The problem statement, all variables and given/known data

    Study the continuity of the following functions, defined by:

    1- f(x) = lim (n^x-n^-x)/(n^x+n^-x) x∈|R
    n->+∞


    2- f(x) = lim [ln(e^n+x^n)]/n x∈|R
    n->+∞

    3. The attempt at a solution

    A function is continuos if its limit L exists and it equals f(L).
    But the limit here is to +∞!
    So, after computing the two limits for the given n->+∞, how do I go on studying the finction?

    Many thanksss
     
  2. jcsd
  3. Nov 17, 2012 #2

    tiny-tim

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    Hi Felafel! :smile:
    You'll get the value of f(x) for various values of x.

    Draw the graph (in your head, if it's easy), and it should be obvious whether it's continuous! :wink:
     
  4. Nov 18, 2012 #3
    thank you :)!
    just.. random values?
     
  5. Nov 18, 2012 #4

    tiny-tim

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    yup! :smile:

    usually works! :biggrin:
     
  6. Nov 18, 2012 #5

    HallsofIvy

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    If you divide both numerator and denominator by [itex]n^x[/itex], you get
    [tex]\frac{1- n^{-2x}}{1+ n^{-2x}}[/tex]
    Now suppose x> 0 and look at three cases, 0< x< 1, x= 1, x> 1.

    Then divide both numerator and denominator by [itex]n^{-x}[/itex] to get
    [tex]\frac{n^{2x}- 1}{n^{2x}+ 1}[/tex]
    And do similary for x< 0.

     
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