Results 1 to 4 of 4

Math Help - Find the limit of the function with the given properties

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    76

    Find the limit of the function with the given properties

    Let f: (0,1) --> R be any function with the following properties.

    1) \lim_{x\to 0}{f(x)} = 0

    2) \lim_{x\to 0}({\frac{f(x)}{x} - \frac{f(x/2)}{x}}) = 0

    Then find: \lim_{x\to 0}{\frac{f(x)}{x}

    I know that f(x) = sin(x^2) follows the above properties. Thus, \lim_{x\to 0}{\frac{f(x)}{x} = 0. But I can't prove this for any general function.

    Any hintss to get me started?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Find the limit of the function with the given properties

    Quote Originally Posted by My Little Pony View Post
    Let f: (0,1) --> R be any function with the following properties.

    1) \lim_{x\to 0}{f(x)} = 0

    2) \lim_{x\to 0}({\frac{f(x)}{x} - \frac{f(x/2)}{x}}) = 0

    Then find: \lim_{x\to 0}{\frac{f(x)}{x}

    I know that f(x) = sin(x^2) follows the above properties. Thus, \lim_{x\to 0}{\frac{f(x)}{x} = 0. But I can't prove this for any general function.

    Any hintss to get me started?
    Applying l'Hopital rule to condition 2) You obtain that \lim _{x \rightarrow 0} f^{'} (x)=0 so that is \lim _{x \rightarrow 0} \frac{f(x)}{x} = \lim _{x \rightarrow 0} f^{'} (x)=0...

    Kind regards

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

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Find the limit of the function with the given properties

    Quote Originally Posted by chisigma View Post
    Applying l'Hopital rule to condition 2)
    We don't know a priori if f is differentiable.
    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

    Re: Find the limit of the function with the given properties

    An alternative that doesn't require the derivative of f(x) starts setting \lim_{x \rightarrow 0} \frac{f(x)}{x}= \lambda. Now setting \frac{x}{2}= \xi we have \lim_{\xi \rightarrow 0} \frac{f(\xi)}{2 \xi}= \frac{\lambda}{2} and the condition 2) becomes...

    \lambda - \frac{\lambda}{2}=0 (1)

    ... and the only value of \lambda that satisfies (1) is \lambda=0...

    Kind regards

    \chi \sigma
    Last edited by chisigma; November 24th 2011 at 01:32 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Find the limit for the given function.
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 28th 2010, 05:17 AM
  2. find a function from the limit
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 20th 2010, 11:48 AM
  3. Find the limit of a function
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: July 1st 2009, 01:35 PM
  4. Replies: 14
    Last Post: April 15th 2009, 05:54 PM
  5. Find the limit for the given function.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 27th 2008, 10:06 PM

Search Tags


/mathhelpforum @mathhelpforum