Results 1 to 4 of 4

Math Help - lim as x --> 0 of [cosec(x)]^x

  1. #1
    Newbie
    Joined
    Jun 2011
    Posts
    1

    lim as x --> 0 of [cosec(x)]^x

    Lim (cosecx)^x
    x-> 0


    how to solve this ? I know the initial step is to take log on both sides, but couldn't proceed further.

    Thanks.


    the answer to the problem is 1
    Last edited by mr fantastic; June 2nd 2011 at 02:44 PM. Reason: Re-titled.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    x\cdot ln(csc(x)) = x\cdot [-ln(sin(x))] = \frac{-ln(sin(x))}{\frac{1}{x}}

    I'm starting to see it.

    There is a reason why you studied trigonometry and algebra. Now is the time to REMEMBER!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by gaganvj View Post
    Lim (cosecx)^x
    x-> 0


    how to solve this ? I know the initial step is to take log on both sides, but couldn't proceed further.

    Thanks.


    the answer to the problem is 1
    limit as x approaches 0 of (Cosec[x])^x - Wolfram|Alpha

    Click on Show steps.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by gaganvj View Post
    Lim (cosecx)^x
    x-> 0


    how to solve this ? I know the initial step is to take log on both sides, but couldn't proceed further.

    Thanks.


    the answer to the problem is 1
    Because is (\sin x)^{-x}= e^{-x\ \ln \sin x} what we have to valuate is...

    \lim_{x \rightarrow 0} x\ \ln \sin x (1)

    From the 'infinite product'...

    \sin x = x\ \prod_{n=1}^{\infty} (1-\frac{x^{2}}{n^{2}\ \pi^{2}}) (2)

    ... we derive...

    x\ \ln \sin x = x\ \{\ln x + \sum_{n=1}^{\infty} \ln (1-\frac{x^{2}}{n^{2}\ \pi^{2}})\} (3)

    ... so that is...

    \lim_{x \rightarrow 0}x\ \ln \sin x =0 (4)

    What is interesting is the fact that the limit (4) is the same for x \rightarrow 0+ and x \rightarrow 0- so that the function (\sin x)^{-x} is continous in x=0...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 3sin(6x)cosec(2x)=4
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: November 2nd 2010, 02:39 PM
  2. LIMxgoesto0(tan(7x)cosec(3x))
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 15th 2010, 03:00 AM
  3. Value of cosec
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 8th 2009, 09:36 AM
  4. cosec(cos^-1 (1/y)
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 11th 2008, 06:11 AM
  5. Sec, Cosec, Cot
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: May 24th 2008, 04:58 AM

Search Tags


/mathhelpforum @mathhelpforum