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Limit prove simple

  1. Nov 20, 2009 #1
    how to prove
    lim(x->a)[tex]\frac{f(x)-f(a)}{x-a}[/tex]=lim(h->0)[tex]\frac{f(a+h)-f(a)}{h}[/tex]

    it seems to be obvious, but i dunno how to prove```
     
  2. jcsd
  3. Nov 20, 2009 #2

    statdad

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    What is the relationship between the denominator [tex] x - a [/tex] and the denominator [tex] h [/tex]?
     
  4. Nov 20, 2009 #3
    Sorry,I have not express clearly.
    what i want to prove is just:
    lim(x->a)[tex](\frac{f(x)-f(a)}{x-a})[/tex]=lim(h->0)[tex](\frac{f(a+h)-f(a)}{h})[/tex]
     
  5. Nov 20, 2009 #4
    Listen to what statdad is saying. What is the relationship by h and x - a? The answer to this question will essentially answer your question.
     
  6. Nov 20, 2009 #5
    use a delta-epsilon argument, the standard delta epsilon definition:

    as x approaches a, we have:

    given [tex]\epsilon[/tex] > 0, there exists a [tex]\delta[/tex] >0 such that for all x with the property 0 < | x - a | < [tex]\delta[/tex], then |f(x) - L | < [tex]\epsilon[/tex].

    In this argument, we have the distance between a point x and a fixed point a bounded between 0 and some fixed [tex]\delta[/tex]. Can you provide a similar argument as h approaches ____ ?
     
  7. Nov 20, 2009 #6
    Why not let h = x - a and rewrite the limit after the substitutions?
     
  8. Nov 21, 2009 #7

    HallsofIvy

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    That is exactly what statdad was suggesting!
     
  9. Nov 21, 2009 #8
    I wasn't sure that that was what statdad was getting at, which is why I posted the equation. Some things are just too subtle, at least for me. :blushing:
     
  10. Nov 22, 2009 #9

    HallsofIvy

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    You still don't get it? In the first limit,
    [tex]\lim_{x\to a}\frac{f(x)- f(a)}{x-a}[/tex]

    let h= x- a. Then x= ??
     
  11. Nov 22, 2009 #10

    Mark44

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    I'm pretty sure none of this is applicable to the problem in this thread.
     
  12. Nov 22, 2009 #11
    No, I understand perfectly how to use h = x - a to transform the first to the second; I just didn't see that statdad was hinting at doing it that way.
     
  13. Nov 22, 2009 #12

    statdad

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    Sorry for any confusion I caused. I've always believed the best horror movies ( and books) are the ones that hint at the source of the horror, and that the best hints are ones that make you puzzle out their meaning. This time, apparently, I was a little too vague.
     
    Last edited: Nov 22, 2009
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