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Understanding limits

  1. Jul 27, 2012 #1
    A limit is the value that a function approaches (without necessarily being equal to) as x approaches a specific value.
    A limit can only exist if the limits approaching from the left and the right both exist and are equal.
    the analogy ive been going off is the idea of a force field or a locked door.
    i think im catching the basic concepts of limits, but getting a little held up on the idea of a limit being discontinuous. it might help if some one could first give a few practical examples of uses of limits in general, and then perhaps also explain the idea of a discontinuous limit please.
     
  2. jcsd
  3. Jul 27, 2012 #2
    When you say "discontinuous limit" are you referring to a function that whose limit exists at a point, say [itex]x_0[/itex] but the function is not continuous at [itex]x_0[/itex]? If so, consider this example:
    [tex]
    f(x) =
    \left\{
    \begin{array}{ll}
    x^2 & \mbox{if } x \neq 0 \\
    100 & \mbox{if } x = 0
    \end{array}
    \right.
    [/tex]

    Now, let's consider [itex]\lim_{x \to 0}f(x)[/itex]. Now, if you are just walking along the graph of this function toward 0 (from either direction), everything will look like this was the graph of the function [itex]g(x) = x^2[/itex], right? I mean, there is no way to tell that the function is not [itex]0[/itex] when [itex]x=0[/itex]. This is a limit that is discontinuous (if I understand what you mean.) Do you see? In a sense, the function doesn't do what it is supposed to do at [itex]x=0[/itex] but it behaves properly at all other points; so, it sort of "tricks" us. So, [itex]\lim_{x \to 0}f(x)=0[/itex] even though [itex]f(0) \neq 0[/itex].


    Also, your definition of limit is slightly wrong. You are referring to the limit of a function. More generally, a limit is the value that some "process" (like a function or a sequence) "approaches" as the argument or index approaches some given value.

    For example, [itex]\lim_{x \to 0}f(x)=0[/itex]. Here, the function is [itex]f[/itex] and the argument is [itex]x[/itex] and it is approaching [itex]0[/itex] and the value of the function is approaching [itex]0[/itex].

    As another example, consider the sequence: [itex] \frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{n}, \dots [/itex]. This sequence is approaching [itex]0[/itex] as [itex]n \to \infty[/itex].
     
    Last edited: Jul 27, 2012
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