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Limit of quotient of two functions

  1. May 24, 2005 #1
    Let f(x) and g(x) be functions.

    Then if limit of f(x)/g(x) = 1. That implies lim f(x) = lim g(x) right?

    Consider this proof.

    lim f(x)/g(x) = 1
    lim f(x) x lim 1/g(x) = 1
    lim f(x) = 1 / (lim 1/g(x))
    lim f(x) = lim g(x).
     
  2. jcsd
  3. May 24, 2005 #2

    arildno

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    No, the implication doesn't follow since the limits of f and g might not exist in the first place at the point where the limit of the quotient is 1.
     
  4. May 24, 2005 #3

    uart

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    Provided that the indivual limits actually exist then yes they will be equal. But just because the limit of f/g exists it doesn't mean that the limits of f and g neccessarily exist.


    EDIT : No I'm not turning into a parrot, I must have posted the same time as arildno. :)
     
    Last edited: May 24, 2005
  5. May 24, 2005 #4

    arildno

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    That's mind-reading, not parroting, uart.
     
  6. May 24, 2005 #5

    uart

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    Or in this case just stating the obvious I think. :)
     
  7. May 24, 2005 #6
    [tex] \lim_{x\rightarrow 1} \frac{\sin x}{x} = 1 [/tex]

    [tex] \lim_{x\rightarrow 1} \sin x \neq 1 [/tex]

    Can a mathematician clarify?
     
  8. May 24, 2005 #7

    arildno

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    Eeh, you've got:
    [tex]\lim_{x\to0}\frac{\sin(x)}{x}=1[/tex]
     
  9. May 24, 2005 #8
    Doh! I retract my previous statement.
     
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