Understanding Limits: Practical Examples & Discontinuity

[Stu21]

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 I've been going off is the idea of a force field or a locked door.
i think I am 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.

 

[Robert1986]

When you say "discontinuous limit" are you referring to a function that whose limit exists at a point, say x_0 but the function is not continuous at x_0? If so, consider this example:
<br /> f(x) =<br /> \left\{<br /> \begin{array}{ll}<br /> x^2 &amp; \mbox{if } x \neq 0 \\<br /> 100 &amp; \mbox{if } x = 0<br /> \end{array}<br /> \right.<br />

Now, let's consider \lim_{x \to 0}f(x). 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 g(x) = x^2, right? I mean, there is no way to tell that the function is not 0 when x=0. 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 x=0 but it behaves properly at all other points; so, it sort of "tricks" us. So, \lim_{x \to 0}f(x)=0 even though f(0) \neq 0. 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, \lim_{x \to 0}f(x)=0. Here, the function is f and the argument is x and it is approaching 0 and the value of the function is approaching 0.

As another example, consider the sequence: \frac{1}{1}, \frac{1}{2}, \ldots, \frac{1}{n}, \dots. This sequence is approaching 0 as n \to \infty.