Results 1 to 2 of 2

Math Help - Where does f'(0) exist?

  1. #1
    Junior Member mremwo's Avatar
    Joined
    Oct 2010
    From
    Tampa, FL
    Posts
    53

    Where does f'(0) exist?

    Question: If r>0 is rational, let f(x):= x^r sin(1/x) and f(0):= 0. For what values of r does f'(0) exists?

    Attempt: So I know this derivative exists when lim(x->0) {[x^r sin(1/x) - 0]/ [x-0]} exists.
    So when lim(x->0) [x^(r-1)sin(1/x)] exists. But I'm a little rusty with limits..

    I was testing out r<1, r=1, and r>1 but then I realized I didn't know these limits. Help would be lovely, thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2

    Arrow

    You got the right idea upto me.
    We want that the following limit to exist
    <br />
\lim_{x\to0}\dfrac{f(x)-f(0)}{x-0}=\lim_{x\to0}x^{r-1}\sin(1/x).<br />
    To compute this lets substitute y:=1/x,
    then x\to0 means |y|\to\infty.
    Hence the limit turns out to be
    \lim_{|y|\to\infty}\dfrac{\sin(y)}{y^{r-1}}=0 if r>1. (bounded/unbounded monotonic function)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Does the limit exist?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 1st 2011, 11:57 PM
  2. Does this function even exist?
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: February 26th 2010, 12:58 AM
  3. does this sum exist?
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: November 26th 2009, 06:53 PM
  4. [SOLVED] sums exist or do not exist
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: November 21st 2009, 10:04 AM
  5. Does f`(0) exist?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 6th 2009, 10:29 AM

Search Tags


/mathhelpforum @mathhelpforum