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Two limit problems

  1. Oct 13, 2006 #1
    Hello,

    I am having difficulties with two limit problems and hopefully someone can show me a trick or two that I don't know!

    1) find [tex]\lim_{x\rightarrow 0}\frac{sin^{2}(\frac{x}{2})}{sin(x)}[/tex]

    Is it just me or is this thing a beast of a limit. I just can't get the denominator to equal anything other then 0!

    I tried using the half angle identity on the top but it simply doesn't help...

    2)find [tex]\lim_{x\rightarrow \frac{\pi}{4}}\frac{1-tan(x)}{sin(x)-cos(x)}[/tex] This one I don't even know where to start, I know that I need to get 1-cos(x)/x or cos(x)-1(x) and/or sin(x)/x or x/sin(x)... but I don't see a way.

    PS- thats x approaches x/2(not pi....) sigh (for q1) and PI/4 (for q2) btw. it's small. :(

    Help! :bugeye:
     
    Last edited: Oct 13, 2006
  2. jcsd
  3. Oct 13, 2006 #2
    For (1) Use L'Hopitals Rule


    For (2) rewrite [tex] 1-\tan x [/tex] as [tex] 1 - \frac{\sin x}{\cos x} [/tex]
     
  4. Oct 13, 2006 #3
    I don't know L'Hopitals Rule. Is it easy to learn? If so I will just look it up and try to learn it :)

    For q2, I will try it, thanks.
     
  5. Oct 13, 2006 #4
    Learned it already, :) well, sorta, :) seems fairly straight forward.
     
  6. Oct 13, 2006 #5

    quasar987

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    I'd be interested in knowing how can 1) be done without l'Hospital. The most natural first step would be to try the substitution

    [tex]\sin^2(\frac{x}{2})=\frac{1-\cos x}{2}[/tex]

    but I don't see where to go from there...
     
  7. Oct 13, 2006 #6
    Yea, it's beastly. I'll ask my teacher and post it :)
     
  8. Oct 13, 2006 #7
    I used l'hospital's for both #1 and #2, :). Very straightforward and immensely useful!
     
  9. Oct 13, 2006 #8

    quasar987

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    mmmh yea but I hope you're aware that if you haven't learned l'Hospital's rule in class yet, and use it in the exam, that's 0 for you. So you'd better learn to do the limits without it.
     
  10. Oct 14, 2006 #9
    In the supplementary note package that came with my course it says that l'hospitals is outside the scope of the course, but says that it can be used. I just never bothered to learn it until now. But I do agree with you 110% that I need to do them both ways if I want to use l'hospitals. Not because I will necessarily lose marks, but because the practice is needed.
    I think I will do a bunch of limit problems since I still can't do #1 without L'Hospitals...
     
    Last edited: Oct 14, 2006
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