Results 1 to 9 of 9

Math Help - Notion of limit

  1. #1
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Notion of limit

    Hello,

    for many years i have been doing derivatives, and its definition: lim f(x+h)-f(x) / h always seemed quite obvious to me, but then i started to think about it...
    h->0

    when we do derivatives through its definition, we usually arrive to something like (h(x+h))/h and we remove the h on the numerator and denominator, leaving x+h, when we replace h by 0 and get to x....but the question is, wont we be doing a violation when we place h by 0? If we condsidered the previous formula, replacing 0 by 0 would cause a 0/0 indetermination, which arises from the fact that the removal rule only applices if h!=0, so....what is that last result we get? The result of ignoring the fact that h cannot be 0 but at the same time assume it is 0?

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member Kanwar245's Avatar
    Joined
    Jun 2011
    From
    Canada
    Posts
    68
    Thanks
    2

    Re: Notion of limit

    Since derivatives are continuous already, we can put h = 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Re: Notion of limit

    Then why did we use a rule that considered that h wont be zero?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1

    Re: Notion of limit

    Quote Originally Posted by Kanwar245 View Post
    Since derivatives are continuous already, we can put h = 0.
    What make you say that. That is false.
    If a function has a derivative at x_0 then the function is continuous at x_0.
    We don't about the derivative at x_0.


    Quote Originally Posted by DarkFalz View Post
    Then why did we use a rule that considered that h wont be zero?
    When evaluating {\displaystyle\lim _{x \to {x_0}}}f(x) we never just let x=x_0.
    That is a fundamental principle of limits. We must look for continuity.
    When considering the difference quotient \dfrac{f(x+h)-f(x)}{h} we hope that the h divides out.
    That can only be if h\ne 0.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Re: Notion of limit

    But after that we place h as 0, so is it valid to place h as 0 or not?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1

    Re: Notion of limit

    Quote Originally Posted by DarkFalz View Post
    But after that we place h as 0, so is it valid to place h as 0 or not?
    If the that means you are not dividing by h.
    What we are really saying, h is approximately zero, but not zero.
    That is a very technical distinction but an important one.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Re: Notion of limit

    If it is not zero, why do we replace h by 0 in the end?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1574
    Awards
    1

    Re: Notion of limit

    Quote Originally Posted by DarkFalz View Post
    If it is not zero, why do we replace h by 0 in the end?
    WE DON'T.
    We simply see what happens for values of h nearly zero.
    Lazy teachers allow replacement. But it really is not correct.
    If the difference quotient reduces to a continuous function of h, then it works.
    It works, but technically not correct.

    Because a derivative is a limit, we must follow the formal definition of limit.
    In the definition of {\displaystyle\lim _{x \to {x_0}}}f(x) it is required that x\ne x_0.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Sep 2011
    Posts
    30

    Re: Notion of limit

    If reducing to a continuous function and replacing works, which is the other way to find the limit?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 18th 2011, 10:00 AM
  2. Notion of Limits and Infinite Series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 22nd 2010, 05:50 PM
  3. Underlined letter in matrix notion
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: May 6th 2010, 03:47 PM
  4. question on notion of 'proves'...
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: April 25th 2010, 12:24 PM
  5. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum