Results 1 to 5 of 5

Math Help - L'Hospital

  1. #1
    Super Member
    Joined
    Jun 2008
    Posts
    829

    L'Hospital

    \lim_{x \to \infty} (\frac{x-4}{x+5})^(x-2)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Apprentice123 View Post
    \lim_{x \to \infty} (\frac{x-4}{x+5})^(x-2)
    Why not show some work

    \ln(y)=\lim_{x\to\infty}(x-2)\ln\left(\frac{x-4}{x+5}\right)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2008
    Posts
    829
    ln \lim_{x \to \infty} y=\lim_{x \to \infty} (x-2).\frac{(x+5)}{(x-4)}.\frac{1}{(x+5)^2}

    ln \lim_{x \to \infty} y=\lim_{x \to \infty} \frac{x^2+5x-2x-10}{x^3+10x^2+25x-4x^2-40x-100}

    ln \lim_{x \to \infty} y=\lim_{x \to \infty} \frac{x^2+3x-10}{x^3+6x^2+15x-100}

    ln \lim_{x \to \infty} y=\lim_{x \to \infty} \frac{2}{6x}

    Its new?


    The answer is: \frac{1}{e^9}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Santa Cruz, CA
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by Apprentice123 View Post
    \lim_{x \to \infty} (\frac{x-4}{x+5})^(x-2)
    Let L=\lim_{x \to \infty} (\frac{x-4}{x+5})^{x-2}

    Taking the natural log of both sides, we have

    \begin{aligned}<br />
\ln(L)&=\lim_{x\to{\infty}}(x-2)\ln\bigg(\frac{x-4}{x+5}\bigg) \\ &=\lim_{x\to{\infty}}\frac{\ln\bigg(\frac{x-4}{x+5}\bigg)}{\frac{1}{x-2}}\end{aligned}

    Applying L'Hopital's Rule, we get

    \lim_{x\to{\infty}}\frac{\frac{x+5}{x-4}\cdot\frac{9}{(x+5)^2}}{-\frac{1}{(x-2)^2}}

    Rewriting, we get:

    \lim_{x\to{\infty}}\frac{-9(x-2)^2}{(x-4)(x+5)}=\color{blue}\boxed{-9}

    But recall that \ln(L)=\lim_{x\to{\infty}}(x-2)\ln\bigg(\frac{x-4}{x+5}\bigg)

    Thus, \ln(L)=-9\implies \color{blue}\boxed{L=e^{-9}}

    I hope that this makes sense!

    --Chris
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2008
    Posts
    829
    yes. Thank you
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. L'Hospital help?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 6th 2010, 06:24 PM
  2. L'Hospital for this?
    Posted in the Calculus Forum
    Replies: 10
    Last Post: November 13th 2008, 12:29 AM
  3. l'hospital
    Posted in the Calculus Forum
    Replies: 7
    Last Post: September 24th 2008, 02:18 PM
  4. L'Hospital
    Posted in the Calculus Forum
    Replies: 22
    Last Post: July 2nd 2008, 02:54 PM
  5. L'Hospital
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 23rd 2008, 03:27 PM

Search Tags


/mathhelpforum @mathhelpforum