Results 1 to 2 of 2

Math Help - L'Hopital's Rule

  1. #1
    Senior Member
    Joined
    Feb 2008
    Posts
    297

    L'Hopital's Rule

    How would I use L'Hopital's Rule to calculate the following limits. I know the rule and can apply it, it's just that, I'm having trouble with some of these questions.

    1)\ \lim_ {x \rightarrow 0^+}\ x^\frac{2}{\ln x}

    2)\ \lim_{x \rightarrow \infty}\ x^ \frac{1}{x}

    2)\ \lim_{x \rightarrow \infty}\ a^ \frac{1}{x},\ a>0
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,964
    Thanks
    1008
    Quote Originally Posted by acevipa View Post
    How would I use L'Hopital's Rule to calculate the following limits. I know the rule and can apply it, it's just that, I'm having trouble with some of these questions.

    1)\ \lim_ {x \rightarrow 0^+}\ x^\frac{2}{\ln x}

    2)\ \lim_{x \rightarrow \infty}\ x^ \frac{1}{x}

    2)\ \lim_{x \rightarrow \infty}\ a^ \frac{1}{x},\ a>0
    For these you will need to use the exponential-logarithmic transformation.

    1) x^{\frac{2}{\ln{x}}}= e^{\ln{\left(x^{\frac{2}{\ln{x}}}\right)}}

     = e^{\left(\frac{2}{\ln{x}}\right)\ln{x}}

     = e^2.


    Therefore \lim_{x \to 0^{+}}x^{\frac{2}{\ln{x}}} = \lim_{x \to 0^{+}}e^2

     = e^2.


    2) x^{\frac{1}{x}} = e^{\ln{\left(x^{\frac{1}{x}}\right)}}

     = e^{\left(\frac{1}{x}\right)\ln{x}}

     = e^{\frac{\ln{x}}{x}}.


    Therefore \lim_{x \to \infty}x^{\frac{1}{x}} = \lim_{x \to \infty}e^{\frac{\ln{x}}{x}}

     = e^{\lim_{x \to \infty}\frac{\ln{x}}{x}}

     = e^{\lim_{x \to \infty} \frac{\left(\frac{1}{x}\right)}{1}} by L'Hospital's Rule

     = e^{\lim_{x \to \infty}\frac{1}{x}}

     = e^0

     = 1.


    3) Surely \lim_{x \to \infty}a^{\frac{1}{x}} = a^0 = 1...
    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, 11:08 AM
  2. L'Hopital's Rule
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 28th 2009, 05:40 PM
  3. líHopitalís Rule Help..
    Posted in the Calculus Forum
    Replies: 5
    Last Post: March 27th 2009, 11:06 PM
  4. L'hopital Rule qn
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 26th 2008, 10:36 AM
  5. L'Hopital Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 8th 2008, 04:00 PM

Search Tags


/mathhelpforum @mathhelpforum