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Limits without L'Hopitals rule

  1. Sep 25, 2005 #1
    how do i find lim x→0 (sin x - tan x)/x³ without using l'hopitals rule?


    also, can someone explain this to me, because I don't understand it.
    given that lim x→c f(x)=L >0. Prove that there exists an open interval (a,b) containing c such that f(x) > 0 for all x ≠ c in (a,b).
     
  2. jcsd
  3. Sep 25, 2005 #2

    Tom Mattson

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    Hello, and welcome to Physics Forums. :smile:

    I've moved this to our Homework section. Please post all homework-type questions here in the future. Also, please see the notice at the top of this Forum.

    I'll pause for you to read it. :biggrin:

    OK, now that you've read the notice, what have you done on this problem?
     
  4. Sep 25, 2005 #3
    for the first one, i've tried to simplify sin x - tan x, but that didn't seem to work out.. and I don't know what else to try.
    and for the second one, i've got no clue where to start.
     
  5. Sep 26, 2005 #4

    lurflurf

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    Trig is your friend
    [tex]\frac{\sin(x)-\tan(x)}{x^3}=-\frac{\sin(x)}{x} \ \left(\frac{\sin\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)}\right)^2 \ \frac{1}{2\cos(x)}[/tex]

    for this
    given that lim x→c f(x)=L >0. Prove that there exists an open interval (a,b) containing c such that f(x) > 0 for all x ≠ c in (a,b)
    The existence of a limit tells us something about a certain open interval
    What is it?
     
  6. Sep 26, 2005 #5

    dextercioby

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    Use the Taylor expansions of the trig functions around 0

    [tex] \sin x\simeq x-\frac{x^{3}}{3!} [/tex]

    [tex] \tan x\simeq x+\frac{x^{3}}{3} [/tex]

    It should come up to [itex] -\frac{1}{2} [/itex].

    Daniel.
     
  7. Sep 26, 2005 #6
    i'm sorry, but i don't how u get from [tex]\frac{\sin(x)-\tan(x)}{x^3}[/tex] to [tex]-\frac{\sin(x)}{x} \ \left(\frac{\sin\left(\frac{x}{2}\right)}{\left(\frac{x}{2}\right)}\right)^2 \ \frac{1}{2\cos(x)}[/tex]
     
  8. Sep 26, 2005 #7
    also for lim x→c f(x)= L >0. prove that there exists and open interal (a,b) containing c such that f(x)>0 for all x ≠ c in (a,b)

    do I just pick out numbers and plug them in? because I don't know what else to do.
     
  9. Sep 26, 2005 #8

    lurflurf

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    If the limit exist then for any h>0 there exist an open interval such that
    |f(x)-L|<h for all x in the open interval
    in particular if L>0 then there exist an open interval such that
    |f(x)-L|<L for all x in the open interval
    or equivalently
    0<f(x)<2L so that
    0<f(x) on some open interval
     
  10. Sep 26, 2005 #9

    lurflurf

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    use
    sin(x)-tan(x)=sin(x)(cos(x)-1)/cos(x)
    cos(x)-1=cos(x)-cos(0)=-2sin((x+0)/2)sin((x-0)/2)=-2(sin(x/2))^2
     
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