Results 1 to 5 of 5

Math Help - L'Hopital's Rule

  1. #1
    Newbie
    Joined
    Jul 2009
    Posts
    12

    L'Hopital's Rule

    Please prove
    lim x-> 0+ (ln cotx)/(e^csc(^2)x) = 0 using L'Hopital's Rule.

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,590
    Thanks
    1445
    Quote Originally Posted by ricey View Post
    Please prove
    lim x-> 0+ (ln cotx)/(e^csc(^2)x) = 0 using L'Hopital's Rule.

    Thanks!
    This is tough, but not impossible.

    We first need to check that this is of indeterminate form.

    It is, since the top \to \infty and the bottom \to \infty.


    So we need to take the derivative of the top and the derivative of the bottom.

    \frac{d}{dx}[\ln{(\cot{x})}] = -\frac{1}{\sin{x}\cos{x}}


    \frac{d}{dx}\left[e^{\csc^2{x}}\right] = -\frac{2\cos{x}e^{\csc^2{x}}}{\sin^3{x}}.


    So \lim_{x \to 0^+}\frac{\ln{(\cot{x})}}{e^{\csc^2{x}}} = \lim_{x \to 0^+}\frac{-\frac{1}{\sin{x}\cos{x}}}{-\frac{2\cos{x}e^{\csc^2{x}}}{\sin^3{x}}}

     = \lim_{x \to 0^+}\frac{\tan^2{x}}{2e^{\csc^2{x}}}


    As x \to 0, \csc^2{x} \to \infty.

    So the bottom \to \infty.


    Therefore this whole limit tends to \frac{0}{\infty} = 0.
    Last edited by Prove It; July 28th 2009 at 05:44 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2009
    Posts
    12
    would deriv of ln cot x be sinx/[(cosx )(-sin ^(2)x]

    = 1/ [(-cosx) (sinx)]
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,590
    Thanks
    1445
    Quote Originally Posted by ricey View Post
    would deriv of ln cot x be sinx/[(cosx )(-sin ^(2)x]

    = 1/ [(-cosx) (sinx)]
    Yes it is.

    I'll fix it now...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2009
    Posts
    12
    okay phew lol
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. L' Hopital's Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 26th 2010, 11:08 AM
  2. líHopitalís Rule Help..
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 27th 2009, 11:06 PM
  3. L'hopital Rule qn
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 26th 2008, 10:36 AM
  4. L'Hopital Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 8th 2008, 04:00 PM
  5. L'Hopital's Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 29th 2008, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum