Calculus 101: What Is a Limit?
- In This Quick Guide:
- What Limit Means
- Can Something Be Nothing?
Calculus teachers are notorious for explaining how to complete a problem (outlining the steps and rules) but not explaining what the problem means. So, in this guide we will look at what a limit actually is before you have to get too nutty with the math part of things.
What Limit Means
Let’s start with a simple function: f(x) = 2x + 5. You know that this is a line with slope 2 and y-intercept 5. If you plug x = 3 into the function, the output will be f(3) = 2 · 3 + 5 = 11. Very simple, everyone understands, everyone’s happy. What else does this mean, however? It means that the point (3,11) belongs to the relation and function I call f. Furthermore, it means that the point (3,11) falls on the graph of f(x), as evidenced in this figure.
The point (3,11) falls on the graph of f.
All of this seems pretty obvious, but let’s change the way we talk just a little to prepare for limits. Notice that as you get closer and closer to x = 3, the height of the graph gets closer and closer to y = 11. In fact, if you plug x = 2.9 into f(x), you get f(2.9) = 2(2.9) + 5 = 10.8. If you plug in x = 2.95, the output is 10.9. Inputs close to 3 give outputs close to 11, and the closer the input is to 3, the closer the output is to 11.
Even if you didn’t know that f(3) = 11 (say for some reason you were forbidden by your evil step-mother, as was Cinderella), you could still figure out what it would probably be by plugging in an insanely close number like 2.99999. I’ll save you the grunt work and tell you that f(2.99999) = 10.99998. It’s pretty obvious that f is headed straight for the point (3,11), and that’s what is meant by a limit.
A limit is the intended height of a function at a given value of x, whether or not the function actually reaches that height at the given x. In the case of f, you know that f does reach the value of 11 when x = 3, but that doesn’t have to be the case for a limit to exist. Remember that a limit is the height a function intends to reach.
Can Something Be Nothing?
You may ask, “How am I supposed to know what a function intends to do? I don’t even know what I intend to do.” Luckily, functions are a little more predictable than people, but more on that later. For now, let’s look at a slightly harder problem involving limits, but before we do, let’s first discuss how a limit is written in calculus.
In our previous example, we determined that the limit, as x approaches 3, of f(x) equals 11, because the function approached a height of 11 as we plugged in x values closer and closer to 3. As it seems with everything else, calculus has a shorthand notation for this:
This is read, “The limit, as x approaches 3, of f(x) equals 11.” The tiny 3 is the number you’re approaching, f(x) is the function in question, and 11 is the intended height of f at 3. Now, let’s look at a slightly more involved example.
The following figure is the graph of 
. Clearly, the domain of g cannot contain x = –2, because that causes 0 in the denominator, and that is just plain yucky.
Notice that the graph of g has a hole at the evil value of x = –2, but that won’t stop us. We’re going to evaluate the limit there. Remember, the function doesn’t actually have to exist at a certain point for a limit to exist—the function only has to have a clear height it intends to reach. Clearly, the function has an intended height it wishes to reach when x = –2 in the graph—there’s a gaping hole at that exact spot, in fact.
The graph of 
How can you evaluate 
? Just like we did in the previous example, you’ll plug in a number insanely close to x = –2, in this case, x = –1.99999. Again, I’ll do the grunt work for you (you can thank me later): g(–1.99999) = –4.99999. As you can see, this function intends to go to a height of –5 on the function g when x = –2. Therefore, 
, even though the point (–2,–5) does not appear on the graph of g(x). This is one example of a limit existing because a function intends to go to a height despite not actually reaching that height.
Now that you understand what a limit is, hopefully the rest of your calculus studies will seem a bit clearer. Good luck!
From The Complete Idiot’s Guide to Calculus, Second Edition, by W. Michael Kelley
