The Physics Philes, lesson 23: Take it to the Limit
In which a new frontier reached, limits are explained, and crappy graphs are drawn.
After unexpectedly taking a week off I am back, refreshed and excited to talk math. I’ve gotten to that point in the book where I am, more or less, completely new to the topic being discussed. It’s kind of scary, but also exciting. Actually, I’m pretty excited about these next couple of weeks of the Physics Philes because I’ll be covering the first thing that tripped me up all those weeks ago: limits.
Limits, it turns out, is a relatively simple concept. It’s just the intended height of a function at a given value. It doesn’t actually matter if the function reaches that height, as long as the function intends to.
If you have a little problem with the idea that a function can have intent, you’re not alone. (Especially if you’ve, say, spent the last five years of your life doing legal things.) But it starts to make sense when you look at some examples. First, let’s get the form down.
As x approaches 3, f(x) equals 11.
OK, so what does that look like in math? I’m glad you asked. It looks a little something like this:
We might say this in words as “the limit, as x approaches 3, of f(x) equals 11. You see that tiny three? That is the number the function is approaching. The function in question is f(x) and 11 is the intended height of f at 3. The weird thing is that the function doesn’t actually need to exist at that point for the limit to exist (we’ll get to when a limit does and does not exist later). The function only has to have a clear height it intends to reach.
If that was all there were to limits, that would be a super easy concept. But of course it has to get a bit more complicated than that. you see, there are things called one-sided limits. You can have a left-hand limit and a right hand limit. Let me try to illustrate this with a crappy picture I drew:
That is pretty terrible. Let me try to explain. A left-hand limit is the height a function intends to reach as you approach x from the left. This is the top curvy line with a hole at (4,6). You can tell if there is a limit at 6 by approaching that point from the left. The left-hand limit written in math looks like this:
Do you see that little minus sign next to the tiny 4? That indicates a left-hand limit. Now, what about a right hand-limit? Well that is just the opposite of the left-hand limit. A right-hand limit is the height a function intends to reach as you approach x from the right. That’s the bottom line on my crappy graph. This is what a right-hand limit looks like in math:
Limits don’t exist everywhere, however. For a limit to exist, three things have to happen.
First, the left-hand limit must exist at x = c. Second, the right-hand limit must exist at x = c. Third, the left- and right-hand limits at c must be equal.
What the heck does this even mean? To find out, you’ll need to look at another one of my crappy drawings.
Look at x = -1 and x = 6. A general limit exists at x = -1, but not at x = 6. Why? Because as you approach -1 from the left and right sides, you’re heading toward y = 5 each time. That means that the one-sided limits exist and are equal, like the requirements dictate. However, if we approach x = 6 from the left and the right we get two different results. When we approach x = 6 from the left side we get y = 5. When we approach x = 6 from the right, we get y = 1. Thus, the right- and left-hand limits are not equal, so a general limit does not exist.
There are also a few ways to know when a limit doesn’t exist. A general limit doesn’t exist if the left-hand and right-hand limits aren’t equal. This stands to reason since it is a requirement that the one-sided limits be equal is explicitly mentioned above. If there is a break in the graph of a function and the two pieces of the function don’t meet at the intended height, there there is no general limit at that point. This plays out in the crappy graph I drew. The graph is split and the two pieces don’t meet at the intended height. According to this rule, the answer played out exactly how it should have.
In addition, a general limit doesn’t exist if a function increases or decreases infinitely at a given x-value. A limit must be a finite number. Do you remember a few weeks ago when we discussed asymptotes? Limits don’t exists where there are asymptotes because the graph goes on and on forever, never reaching a finite number.
Finally, a general limit does not exist if the function oscillates infinitely, never approaching a single height. This is super rare.
There are a couple of important things to remember. If a graph has no general limit at one of its x-values, that doesn’t affect any of the other x-values. Giving a limit answer of ∞ or -∞ is tantamount to saying that a limit doesn’t exist. This explains why the limit doesn’t exist and whether the function increased or decreased into infinity at that point.
So these are limits. Stay tuned next week when I learn how to numerically evaluate limits. Should be a raucous thrill ride.
Featured image credit: the mad LOLscientists