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Extra credit Abstract limit question

  1. Apr 7, 2012 #1
    1. The problem statement, all variables and given/known data

    Let f be a derivable function at 0 and f'(0)=2 and let a and b in ℝ.

    Calculate the limit: [tex]\lim_{x\rightarrow0}\frac{f(ax)-f(bx)}{x}[/tex]


    3. The attempt at a solution

    I'm not sure but i got 2a-b as my answer but i wan't to know how to solve it the proper way any help is very much appreciated.
     
  2. jcsd
  3. Apr 7, 2012 #2

    SammyS

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    2a-b is not the correct answer. 2(a-b), or a-b may possibly be the answer.

    Write f ' (0) as a limit & see where you need to go from there.
     
  4. Apr 7, 2012 #3
    The limit is of indeterminate form [tex]\frac{0}{0}[/tex] so you can apply L'Hopital's rule to it to get the limit
    [tex]\lim_{x\rightarrow0}\frac{af'(ax)-af'(bx)}{1}[/tex]

    which since we know [tex]f'(0)=2[/tex]

    should be easy to evaluate.
     
  5. Apr 7, 2012 #4

    Dick

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    You could also do it directly. Add and subtract f(0) in the numerator, split it up and apply changes of variable like u=ax and v=bx.
     
  6. Apr 7, 2012 #5

    SammyS

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    That's not what I intended.

    Use:
    [itex]\displaystyle f'(0)=\lim_{h\to0}\ \frac{f(0+h)-f(0)}{h}[/itex]
    Of course that's the same as [itex]\displaystyle f'(0)=\lim_{x\to0}\ \frac{f(x)-f(0)}{x}\ .[/itex]

    Now try using that in the limit you're trying to evaluate. (In the way Dick mentioned.)
     
  7. Apr 7, 2012 #6
    Wow thanks a lot i didn't realize it was that simple :).
     
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