You won't understand this post if you don't already know algebra ideas, however. Sorry! We'll get there eventually and then you can come back to it.
So, what are limits?
The most basic explanation is that a limit is the behaviour of a function as it approaches a certain value. What does f(x) do as x approaches infinity or zero or 4?
Why is that important? Why do we need to know if f(x) is going to infinity or whatever? First of all, continuity. Secondly, the definition of derivatives goes back to limits, and FURTHERMORE, since differential equations model actual real-life scenarios, it can tell us what to expect the outcomes of those scenarios to be.
The last two are outside the scope of an introduction to limits, ESPECIALLY if I'm framing this at an algebra level, so let's talk about continuity.
"Dudes already know about continuity!!"
Whatever I'm explaining it anyway. Continuity basically means there are no holes in the graph of a function. If you put your pencil down at the far left side of the graph, you can get to the far right side of the graph without ever lifting it.
The mathematical definition of continuity at any point is:
This actually says a few different things:
1. The limit exists, which means:
2. The limit from the left and the limit from the right are equal to each other
3. The limit as x approaches a is equal to the value of the function at x = a
Let's do this with a simple function first to give you an example of what I'm talking about. Here's the graph of F(x) = 2x+1:
It should be easy enough to see that this graph is continuous. F(x) will exist for any possible x you might pick, anywhere from -∞ to +∞. So how does that fit in with our definition of continuity?
Let's try a limit with a at specific point. What does the function do as x approaches 0? In this case, we can just plug in 0 to see what happens:
The limit exists at that point, and it is equal to the function at that point, which means F(x) is continuous at x = 0. Pretty easy!
What happens when you pick a number you can't plug in, though? Like, say, INFINITY?
Infinity's not a number. It's an idea. You can't plug infinity into 2x+1 to see what you get. What you CAN do is look at the graph to figure out what it's doing as x gets infinitely larger. In this case, as x increases, F(x) is also increasing. It will keep going up and up and up on that cartesian plane FOREVER because there is nothing to stop it. Thus, the limit is also going to be infinity:
That should pretty much be self-evident. It doesn't MEAN a goddamn thing for this particular function, sadly, but it's an example of how to take limits to infinity.
Let's try it for a function that's NOT continuous for all values of x.
This graph has an asymptote through x = 1. Let's look at how that discontinuity works with the graph's limit.
We can't just plug in 1 here because HOLY SHIT DIVISION BY 0 so don't do that you'll wreck the universe or something. (spoilers: I'm going to blow your mind later by dividing by zero.)
What you have to do in this case is take the limit from both sides. If the limit from the left and the limit from the right are equal to each other, then the limit exists. If they're not equal, then the limit does not exist, which means you can write a big fat DNE on your assignment.
Remember that whole number line thing? Imagine 1 on a number line. Now imagine you're running at it from the positive direction: starting at the rightmost side of the number line and sprinting toward 1. We'd write that as:
How do you figure that out? Plug in a bunch of numbers that are greater than 1 but closer and closer to it. Tables are very helpful for this:
| x | F(x) |
| 1.9 | 1.1111 |
| 1.09 | 11.1111 |
| 1.009 | 111.1111 |
As you can see, the closer we get to 1 from the right, the bigger F(x) is getting. You can keep plugging in values for x until you're satisfied, but I'm gonna say this limit is +∞ and call it a day.
Now we do the same thing for 1 from the left. Start running toward 1 from the negative direction and see what happens.
| x | F(x) |
| .009 | -10 |
| .09 | -100 |
| .9 | -1000 |
Looks like this one's going to -∞ the closer you get to 1. So it's positive infinity from the right, but negative infinity from the left, which means...
Whoops! Limit DNE!
Since the limit at 1 doesn't exist, because the limits at 1+ and 1- aren't equal, that means the function is discontinuous at x=1.
But there's two different ways to make a function discontinuous according to our definition. We've only proven the first way: if the limit doesn't exist. Let's take a look at the second way: if the limit as x approaches a is not the same as F(a).
We'll use a removable discontinuity for this one. That means you can rewrite it as a continuous function with algebraic manipulation.
This can be re-written as:
Since you can factor an x - 1 from the top and cancel it with the bottom, that means the function is continuous, right? Well, let's take a look:
So our limit is 2, right? Right! The limit exists and it is 2 as x goes to 1. In order for the function to be continuous at x = 1, however, the original function must also equal 2 at x = 1.
Whoooops division by zero! F(1) doesn't exist!
BUT WAIT! Why can't you just cancel it like you did before, Guindo?
That's actually a very good question, and it is a common error. Let's take a look at why that doesn't work for this particular value of x.
For any non-one value of x, it's true that
You still can't divide by zero. When you cancel a fraction like this, what you're really saying is that its value is 1 so continuing to write it would be frivolous. But
So the graph has a hole in it at x=1, and the function is discontinuous.
Gosh this limits post sure got long, didn't it? We've sufficiently covered continuity, at least; I can talk about calculating tricky limits and indeterminate forms next time.
Oh, wait, there is one trick that math books like to throw you about continuity: using piecewise functions to ask if something's continuous. This is actually really easy to tackle if you keep the definition of continuity in mind.
I'm going to use our last example for this one, since we already did the work of finding the limit, and give it a specific condition as a piecewise function:
If x is NOT 1, then the top line is what we graph. If x IS 1, then the bottom line is what we graph. In a case like this, what the writer of the problem is really asking you to do is take the limit as x approaches the value where the piecewise function changes, and then make sure the value of the function itself is equal to the limit at that point.
We've already done this, so we know that the limit as x approaches 1 is 2, and since the piecewise function says that F(1) = 2, the function is continuous at that point. Math books do this to try to trip you up, but if you have an understanding of continuity based on limits, it's just a cheap trick.
Next time: Indeterminant forms and fun ways to bend the universe around limits
* Graphs supplied by this fine website. Code...well, that's in the sidebar now, isn't it?
** Why can we cancel (x-1)/(x-1) for the limit but not when we plug it into the function? When you take a limit, you're only looking at it's behaviour as x GETS CLOSE TO a. You're not actually taking its value AT a. That means a limit as x approaches 1 will never have x EQUAL 1, so (x-1)/(x-1) will never equal 0/0.
