Results 1 to 3 of 3

Math Help - Limit proof

  1. #1
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Limit proof

    Hi, I've to proof that:

    \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=1

    This is what I found:

    \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=\lim_{x \to 0} \left(\int_{x}^{\frac{\pi}{2}} \frac{du}{\sin^2(u)} - \int_{x}^{\frac{\pi}{2}}\frac{\cos(u)du}{\sin^2(u)  }\right)
    =\lim_{x \to 0} \left( \left[-\cot(u)\right]_{x}^{\frac{\pi}{2}}+\left[\frac{1}{\sin(u)}\right]_{x}^{\frac{\pi}{2}}\right)
    =\lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)

    The problem is here, because \lim_{x \to 0} \cot(x)=+\infty and \lim_{x \to 0} \frac{1}{\sin(x)}=\infty} I don't now how to go further.

    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3

    Re: Limit proof

    Quote Originally Posted by Siron View Post
    Hi, I've to proof that:

    \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=1

    This is what I found:

    \lim_{ x \to 0} \int_{x}^{\frac{\pi}{2}} \frac{1-\cos(u)}{\sin^2(u)}du=\lim_{x \to 0} \left(\int_{x}^{\frac{\pi}{2}} \frac{du}{\sin^2(u)} - \int_{x}^{\frac{\pi}{2}}\frac{\cos(u)du}{\sin^2(u)  }\right)
    =\lim_{x \to 0} \left( \left[-\cot(u)\right]_{x}^{\frac{\pi}{2}}+\left[\frac{1}{\sin(u)}\right]_{x}^{\frac{\pi}{2}}\right)
    =\lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)

    The problem is here, because \lim_{x \to 0} \cot(x)=+\infty and \lim_{x \to 0} \frac{1}{\sin(x)}=\infty} I don't now how to go further.

    Thanks in advance!
    Hi Siron,

    \lim_{x \to 0} \left(\cot(x)+1-\frac{1}{\sin(x)}\right)=\lim_{x \to 0} \left(\frac{\cos x+\sin x-1}{\sin x}\right)

    Since this limit is in a indeterminate form I suggest you use the L'Hopital's rule.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Limit proof

    Thanks!
    I've solved it now.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] limit proof
    Posted in the Differential Geometry Forum
    Replies: 18
    Last Post: December 23rd 2010, 09:48 PM
  2. Limit Proof
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 15th 2009, 02:33 AM
  3. Limit Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 5th 2008, 04:17 AM
  4. limit proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 4th 2008, 06:27 AM
  5. Limit Proof
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 8th 2006, 03:09 PM

Search Tags


/mathhelpforum @mathhelpforum