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Limit times function

  1. Oct 8, 2014 #1
    Is the following true, if no is there som theory i can studdy?

    ##\lim_{\Delta x \to 0}f(x+\Delta x) \cdot \Delta x = 0##
     
  2. jcsd
  3. Oct 8, 2014 #2
    This is not always true. It is true if ##f## is continuous at ##x## or if the ##\lim_{\Delta x \to 0}f(x+\Delta x) ## exists; use the product law for limits. It is true if ##f## is bounded near ##x##; use the Squeeze Theorem.

    If ##f## is unbounded near ##x##, the the limit may exist but not be ##0##, or it just might fail to exist. Look at ##f(x)=1/x## and ##f(x)=1/x^2## with ##x=0##.
     
  4. Oct 8, 2014 #3
    Thanks, but i really need to show the following if its possibole.

    ## \lim_{\Delta x \to 0} f(x_0 + \Delta x) \cdot \Delta x= k ## Where k, is a constant.

    Is there someting i can say aboute f?
     
  5. Oct 8, 2014 #4

    HallsofIvy

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    What do you mean by "possible"? You have already been told that it is NOT true in general. You have also been told that if f is continuous at [itex]x_0[/itex] or if [itex]\lim_{\Delta x\to 0} f(x+ \Delta x)[/itex] exits then it is true.
     
  6. Oct 8, 2014 #5
    I was told in general, so there must be som functions that does the oppeist?
     
  7. Oct 9, 2014 #6

    HallsofIvy

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    Who told you that? gopher_p, in the only other response here, said "this is not always true".

    Take f(x)= 1/x, [itex]x_0= 0[/itex].
     
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