Results 1 to 3 of 3

Math Help - l'Hôpital's Rule

  1. #1
    siriano.3
    Guest

    l'Hôpital's Rule

    lim [ln(x^4-3)]/[ln(x)cos(1/x)]
    x-> INF

    So, that is the form INF/INF, so I tried to take the derivative of the numerator and denominator, but I can't figure out the derivative of the denominator.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by siriano.3 View Post
    lim [ln(x^4-3)]/[ln(x)cos(1/x)]
    x-> INF

    So, that is the form INF/INF, so I tried to take the derivative of the numerator and denominator, but I can't figure out the derivative of the denominator.
    it's a doozie!

    by the product rule and chain rule:

    \frac d{dx} \ln x \cos \left( \frac 1x \right) = \frac 1x \cos \left( \frac 1x \right) + \frac {\ln x}{x^2} \sin \left( \frac 1x \right)

    try seeing what you can simplify before applying L'Hopital's
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    \lim_{x\to \infty} \frac{\ln (x^4-3)}{\ln x \cos \frac{1}{x}} since \lim_{x\to \infty} \frac{1}{\cos \frac{1}{x}} = 1. This problem reduces to:
    \lim_{x\to \infty} \frac{\ln (x^4-3)}{\ln x}.
    We can solve this without L'Hopital, note when x is large:
    \ln (x^4) - \ln (3) \leq \ln (x^4 -3) \leq \ln (x^4)
    Thus,
    \frac{\ln x^4 - \ln 3}{\ln x} \leq \frac{\ln (x^4-3)}{\ln x} \leq \frac{\ln x^4}{\ln x}.
    Now the left side and right side limits are 4, because, for example,
    \lim_{x\to \infty} \frac{\ln x^4 - \ln 3}{\ln x} = \lim_{x\to \infty} \frac{\ln x^4}{\ln x} - \underbrace{\frac{\ln 3}{\ln x}}_{\mbox{ this is zero }}.
    But,
    \lim_{x\to \infty} \frac{\ln x^4}{\ln x} = \lim_{x\to \infty} \frac{4\ln x}{\ln x} = 4.
    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, 12:08 PM
  2. L'Hopital's Rule
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 28th 2009, 06:40 PM
  3. l’Hopital’s Rule Help..
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 28th 2009, 12:06 AM
  4. L'hopital Rule qn
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 26th 2008, 11:36 AM
  5. L'Hopital Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 8th 2008, 05:00 PM

/mathhelpforum @mathhelpforum