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Help to Prove f continuous

  1. Mar 12, 2012 #1
    1. The problem statement, all variables and given/known data
    Define f:[-1,∞]→ℝ as follows: f(0) = 1/2 and

    f(x) =[(1 + x)^(1/2) - 1]/x , if x ≠ 0

    Show that f is continuous at 0.


    2. Relevant equations
    Definition. f is continuous at xo if xoan element of domain and
    lf(x) - f(xo)l < ε whenever lx - xol < δ


    3. The attempt at a solution
    Do some algebra come up with f(x) = 1/[(1+x)^(1/2) + 1]

    I also know that between [-1,1], f(x)≤ 1
    and x ≥ 1, f(x) ≤ 1

    I know I need to somehow pull out an lxl from the absolute value, since I know lxl≤δ then
    I can define δ in terms of ε and the function.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 12, 2012 #2

    SammyS

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    Do you need to use an ε - δ proof, or can you use the limit criterion for continuity?
    If [itex]\displaystyle \lim_{x\to x_0}\,f(x)=f(x_0)\,,[/itex] then f is continuous at x0 .​
     
  4. Mar 12, 2012 #3
    It just says to show, it doesn't specify δ-ε proof. I've spent hours working on this one problem, any suggestions greatly appreciated!
     
  5. Mar 12, 2012 #4

    SammyS

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    What is the following limit?
    [itex]\displaystyle
    \lim_{x\to0}\,\frac{1}{(1+x)^{1/2} + 1}
    [/itex]​
     
  6. Mar 12, 2012 #5
    It is 1/2 which is f(0).
    So this approach I show: 1) the point c is in the domain
    2) the limit of f(c) exists and
    3) lim x->c f(x)=f(c)

    I should have thought of this, it is a lot easier to show. Thank you.
     
  7. Mar 12, 2012 #6

    SammyS

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    You're welcome!
     
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