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A question about the derivative

  1. Oct 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Generally the derivative has the limit x-- h applied to the whole thing like
    $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

    I'm guessing you can't express it as

    $$\frac{\lim_{h\to 0} f(x+h)-f(x)}{\lim_{h\to 0} h}$$

    because the quotient rule for limits doesn't hold when the limit of bottom part of the fraction equals 0.

    Can you express it like this though?

    $$\frac{f(x +\lim_{h\to 0} h)-f(x)}{\lim_{h\to 0} h}$$

    In other words, does $$\lim_{h\to 0} \frac{f(x+h)-f(x)}{h} = \frac{f(x +\lim_{h\to 0} h)-f(x)}{\lim_{h\to 0} h}$$ ?
     
  2. jcsd
  3. Oct 26, 2012 #2

    SammyS

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    The limit of the denominator is zero in that expression too. So, no, you can't do that.
     
  4. Oct 26, 2012 #3

    Mark44

    Staff: Mentor

    Besides what SammyS said, you can't in general "distribute" the limit operation into a function.

    IOW, it's generally not true that
    ## \lim_{h \to 0} f(x + h) = f(x + \lim_{h \to 0} h)##
     
  5. Oct 26, 2012 #4
    Your latter formula means the denominator and the numerator are not synchronized when h--->0
    Could you understand me?
     
  6. Oct 27, 2012 #5

    Mark44

    Staff: Mentor

    Who is this directed to, and what do you mean?
     
  7. Oct 27, 2012 #6
    Thanks, Mark44 and SammyS :)
    Bennett.F.L, I'm sorry, I'm not sure what you mean. It's ok though, I think I got it.
     
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