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Math Help - L'Hopital's Rule

  1. #1
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    L'Hopital's Rule

    I have to use l'hopital's rule to find each limit:

    1. lim x approaching 0 from the right of

    (1-cos(sqrt x))/x

    Since this is a 0/0 form I use l'hopitals rule. making it:

    Lim x approaching 0 from the right of (sin(sqrt x))/(2 sqrt x).

    Again this is a 0/0 form but if I just keep applying l'hopital's rule it always comes up with a 0/0 form. How do I solve this?
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  2. #2
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    You just need to apply it once more.

    (sin(sqrt x))/(2 sqrt x) -> 1/2x^(-1/2)sin(sqrt x) / x^(-1/2) simplifies to 1/2cos (sqrt x).

    Forgive me if the details aren't right, but I'm pretty sure the idea is there.
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  3. #3
    MHF Contributor matheagle's Avatar
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    I'd let x=u^2 which makes this much easier to work with

     \lim_{u\to 0^+}{1-\cos u\over u^2} ={1\over 2}\lim_{u\to 0^+}{\sin u\over u} ={1\over 2}
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