Results 1 to 8 of 8

Math Help - These crazy limits..

  1. #1
    Member
    Joined
    Apr 2010
    Posts
    81

    These crazy limits..

    I understand how to work with basic limits, and those ones under square roots, & how to implement that sinx/x=1 identity (& cos, tan).

    But now, I'm coming across some really confusing stuff & my teacher simply is incapable of expressing these in intelligible terms.

    here are some examples:
    <br />
\displaystyle \lim_{x \to 0} \frac{ x^2 }{ \cos 8 x - \cos 4 x } =<br />
    <br />
\displaystyle \lim_{x \to 1^{+}} ( \displaystyle \frac {1}{\ln x} - \displaystyle \frac {1}{x - 1} ) =<br />
    <br />
\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =<br />
    <br />
\displaystyle \lim_{x\to \pi^+} \left(4x-4\pi \right)\tan\!\left(\frac{x}{2}\right)<br />

    I'm not asking for answers to all of these. But can some one point me to some videos/material that can teach me how to solve & understand these particular types of limits? I have no idea where to begin!

    Perhaps a sample calculation for just the first one too.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Oct 2010
    Posts
    127
    I'll give you a heads up for the first one. This is the url. Good luck. These problems tend to be long. But you can do it.

    http://i52.tinypic.com/4pz8zp.jpg
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Apr 2010
    Posts
    81
    Thanks!

    Is L'Hopital's rule needed for all of these?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Vamz View Post
    Thanks!

    Is L'Hopital's rule needed for all of these?
    Well, can they be put into a form where l'Hopspital's Rule can be used?

    The third one can be done simply by making a simple substitution and using a well known limit definition for e^x ....


    Moderator edit: The third question was double posted (against forum rules, by the way) and has been further discussed here: http://www.mathhelpforum.com/math-he...tml#post582776.
    Last edited by mr fantastic; November 11th 2010 at 06:13 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Vamz View Post
    I understand how to work with basic limits, and those ones under square roots, & how to implement that sinx/x=1 identity (& cos, tan).

    But now, I'm coming across some really confusing stuff & my teacher simply is incapable of expressing these in intelligible terms.

    here are some examples:
    <br />
\displaystyle \lim_{x \to 0} \frac{ x^2 }{ \cos 8 x - \cos 4 x } =<br />
    <br />
\displaystyle \lim_{x \to 1^{+}} ( \displaystyle \frac {1}{\ln x} - \displaystyle \frac {1}{x - 1} ) =<br />
    <br />
\displaystyle \lim_{x \to + \infty} \left( 1 + \frac {3}{x} \right)^{x/2} =<br />
    <br />
\displaystyle \lim_{x\to \pi^+} \left(4x-4\pi \right)\tan\!\left(\frac{x}{2}\right)<br />

    I'm not asking for answers to all of these. But can some one point me to some videos/material that can teach me how to solve & understand these particular types of limits? I have no idea where to begin!

    Perhaps a sample calculation for just the first one too.

    Thanks
    If you're interested, for my senior High School project I wrote a 100 page manual on how to compute difficult limits, I could send it to you. Besides that I don't know many places where you can find material on "techniques" of limit evaluation.

    Here's a thread you might find interesting from my past life
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Apr 2010
    Posts
    81

    Limit regarding pi

    <br />
\displaystyle \lim_{x\to \pi^+} \left(4x-4\pi \right)\tan\!\left(\frac{x}{2}\right)<br />


    <br />
\displaystyle let f(x)=(4x-4\pi) ; g(x)=\tan(\frac{x}{2})<br />

    so, therefore
    <br />
\displaystyle Lim( f(x)* g(x) ) = lim F(x) * lim G(x)<br />

    for g(x)
    <br />
\displaystyle U=\frac{x}{2}<br />
    <br />
\displaystyle U*\frac{\tan U}{U} = U * 1 = U<br />

    So, the limit for g(x) is \displaystyle\frac{\pi}{2}

    Now, for dealing with f(x)
    <br />
\displaystyle (4x-4\pi) =0<br />

    no matter how I move these variables around, f(x) always ends up zero, sending my entire limit to zero - which is incorrect! What am I missing here?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Chicago, IL
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by Vamz View Post
    <br />
\displaystyle \lim_{x\to \pi^+} \left(4x-4\pi \right)\tan\!\left(\frac{x}{2}\right)<br />

    <br />
\displaystyle let f(x)=(4x-4\pi) ; g(x)=\tan(\frac{x}{2})<br />

    so, therefore
    <br />
\displaystyle Lim( f(x)* g(x) ) = Lim f(x) * Lim g(x)<br />

    for g(x)
    <br />
\displaystyle U=\frac{x}{2}<br />
    <br />
\displaystyle U*\frac{\tan U}{U} = U * 1 = U<br />

    So, the limit for g(x) is \displaystyle\frac{\pi}{2}

    Now, for dealing with f(x)
    <br />
\displaystyle (4x-4\pi) =0<br />

    no matter how I move these variables around, f(x) always ends up zero, sending my entire limit to zero - which is incorrect! What am I missing here?

    Thanks!
    \lim\limits_{x\to\pi^+}\tan\left(\frac{x}{2}\right  )=-\infty. But \lim\limits_{x\to\pi^+}(4x-4\pi)\tan\left(\frac{x}{2}\right)\to 0\cdot -\infty\neq 0!!!

    Thus, rewrite the limit as follows:

    \lim\limits_{x\to\pi^+}\dfrac{4x-4\pi}{\cot\left(\frac{x}{2}\right)}\rightarrow \dfrac{0}{0}

    At this stage, now apply L'Hopitals rule. Can you proceed?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Apr 2010
    Posts
    81
    Agh, thats awesome. Thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Going crazy on this log question!
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: October 16th 2009, 05:35 AM
  2. Crazy problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: June 12th 2009, 02:26 PM
  3. crazy limit
    Posted in the Calculus Forum
    Replies: 22
    Last Post: June 13th 2008, 08:00 AM
  4. Trig Crazy
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: October 2nd 2007, 10:23 PM
  5. I am going crazy, please help!
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 22nd 2006, 07:15 AM

Search Tags


/mathhelpforum @mathhelpforum