Let's try to understand the concepts of limits and continuity with an intuitive approach. In this page I'll introduce briefly the ideas behind these concepts. These ideas are explored more deeply in the links below.
When we talked about functions before, we payed attention at the values of functions at specific points. For example, the value of f(x) at x=1.
The idea behind limits is to analyze what the function is "approaching" when x "approaches" a specific value. To start getting used to this idea, let's turn to this graph:
When x approaches the value "a" in the x axis, the function f(x) approaches "L" in the y axis. In this graph I drawed a big pink hole at the point (a,L). I do this because we don't necessarily know the value of function f at x=a.
Let's turn to the graph of a function whose expression we know:
This is read "the limit as x approaches 1 of x squared equals 1".
You might ask what this is useful for. Very good question. Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right?
Well, the point is that sometimes we don't care what the function is at x=1.
As an example of this, let's consider the following function:
What does the function approach when x approaches 1? It also approaches 1, right? It doesn't matter that the function is other than 1 at that point! So,
In calculus, the most useful limits are like this one. The value of the function at the specific point we care about is not defined, like 0/0 (which is complete junk), or useless, like zero or infinite.
In these cases we can know what the function is approaching, and that is what we really need. To learn more:
Limits and continuity are often covered in the same chapter of textbooks. This is because they are very related. The basic idea of continuity is very simple, and the "formal" definition uses limits.
Basically, we say a function is continuous when you can graph it without lifting your pencil from the paper. Here's an example of what a continuous function looks like:
There is a precise mathematical definition of continuity that uses limits, and I talk about that at continuous functions page. Intuitively, this definition says that small changes in the input of the function result in small changes in the output.
If you are confused by that, ignore it! You don't need to learn all at once. The most important is to recognize a continuous function when you see it.
Now, what would a discontinuous function look like? A function essentially is discontinuous when it has any "gap". For example:
Easy to recognize, isn't it?
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