Results 1 to 12 of 12

Math Help - Functions without limits

  1. #1
    Banned
    Joined
    Oct 2009
    Posts
    56

    Functions without limits

    What does it mean to say that f(3) exists but that there is no limit to f(x) as x approaches 3?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    It means that the limit is dependent upon the path you take.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Random Variable View Post
    It means that the limit is dependent upon the path you take.
    Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    56
    Quote Originally Posted by Drexel28 View Post
    Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}
    So there's no limit on that one because 1/|x-3| converges on infinitesimality?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by satx View Post
    So there's no limit on that one because 1/|x-3| converges on infinitesimality?
    Did you just say infinitesimality? haha. It diverges...thus it does not exist.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2009
    Posts
    56
    Quote Originally Posted by Drexel28 View Post
    Did you just say infinitesimality? haha.
    Don't hate; it's a perfectly cromulent word
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,965
    Thanks
    1785
    Awards
    1
    Quote Originally Posted by satx View Post
    What does it mean to say that f(3) exists but that there is no limit to f(x) as x approaches 3?
    f(x) = \left\{ {\begin{array}{rl}   {x~~,} & {x \geqslant 3}  \\   {x - 1,} & {x < 3}  \\ \end{array} } \right.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    Quote Originally Posted by Drexel28 View Post
    Not always. Trying to list all the ways in which a limit can fail to exist is a tough task. What about f(x)=\begin{cases} 0 &\mbox{if} \quad x=3\\ \frac{1}{|x-3|} & \mbox{if} \quad x\ne 3\end{cases}
    Obviously if the function increases (or decreases) without bound then the limit doesn't exist. But what's another way?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    What about a function that oscillates wildly near a point?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Random Variable View Post
    What about a function that oscillates wildly near a point?
    Indeed another. What about the function not even existing on an open ball around the point while we're going here.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Super Member Random Variable's Avatar
    Joined
    May 2009
    Posts
    959
    Thanks
    3
    Quote Originally Posted by Drexel28 View Post
    Indeed another. What about the function not even existing on an open ball around the point while we're going here.
    Does that count?

    If I remember correctly, in real analysis we proved that  \lim_{x \to 0} \sin(1/x) does not exist by finding a sequence (s_n) that converges to some value c (  s_{n} \ne c) but where f(s_n) doesn't converge (or something like that).
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by Random Variable View Post
    Does that count?

    If I remember correctly, in real analysis we proved that  \lim_{x \to 0} \sin(1/x) does not exist by finding a sequence (s_n) that converges to some value c (  s_{n} \ne c) but where f(s_n) doesn't converge (or something like that).
    Taking a_n=\frac{1}{2\pi n+\frac{\pi}{2}},b_n=\frac{1}{2\pi n}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Limits of Functions
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: October 17th 2010, 01:53 PM
  2. Limits of functions
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: April 26th 2010, 08:41 PM
  3. Functions and limits?
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 10th 2009, 02:07 PM
  4. limits of functions
    Posted in the Calculus Forum
    Replies: 5
    Last Post: February 16th 2009, 12:35 PM
  5. Limits of e functions
    Posted in the Calculus Forum
    Replies: 9
    Last Post: September 29th 2008, 12:20 PM

Search Tags


/mathhelpforum @mathhelpforum