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:
$$f(x) = \left\{ \begin{array}{ll} x^2 & \mbox{if } x \neq 0 \\ 100 & \mbox{if } x = 0 \end{array} \right.$$

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 $n \to \infty$.