The delta epsilon proof is also known as the Precise Definition of a Limit
. To most eyes, however, it looks like a bunch of absolute gibberish
until it's translated into English. And even then, the mathematical
stuff will continue to look like gibberish until you really hunker down and try to understand the thing, matching all the strangely named variables
to real life things, or at least graphs.
This node assumes you know what a function is, and how to plot one on a graph.
The delta epsilon proof, in its purest mathematical form, looks like this:
∀ε>0 ∃δ>0 ∋ 0<|x-a|<δ⇒|ƒ(x)-L|<ε
Whew! That looks downright scary, it does! Here's a list of the weird symbols involved, and what they mean in English, or at least, what they mean in half-English/half-Mathmatics.
∀ "For all" or "For every". This means that the next symbol is going to be a variable, and that you should consider that variable to equal ANY arbitrary number, not just one that you'd like to pick at this moment. Sure, you can pick a number and plug it in if you like, but you still need to show that it works with ANY number, not just that one. (in this example, there is one additional hidden restriction on the number you can pick, but that'll be discussed later)
ε "Lower case epsilon" This is just a variable. You know, like x or y. Except instead of using calm, soothing letters we're familiar with, they chose dark, scary, eldritch Greek letters that no one but "scientists" ever use anymore. Forget the scary part, this is just a variable.
> Greater than. You should have learned this one in elementary school (or primary school, or whatever your country offers).
The number zero, indicating emptiness, nothingness, and the origin point of the number line. Less than zero is debt, greater than zero is having something.
∃ "There exists". That's pretty much what this means to it's fullest extent.
δ "Lower case delta" Here's another variable, also a Greek letter, but no different than any other variable you've ever come across except that it uses a symbol you aren't yet familiar with. Deal with math or physics enough though, and you'll get familiar quickly. Yes that's a threat. Deal with it and get over yourself. This isn't hard!
∋ "Contains as a member" is the official meaning of this symbol, but my calculus professor explained this one as meaning "Such that". And in terms of this particular line of gibberish, that's the most accurate translation, so that's what I'll be referring to it as. Brontosaurus informs me that my professor's use of this symbol here is non-standard, and that he usually sees either a comma, a single vertical line "|", or three dots "∴". Since I'm basically copying from my notes here, I'm going to leave it as my professor wrote, but keep in mind that you'll probably never see it like this anywhere else in the world.
< "Less than". Like greater than, you should know this one already.
| This is the "absolute value" sign. While in some situations, it can take on the role of a parenthesis, it means more than that. Technically, this means "how far away from the origin are you" but in most situations, all it means is that if the stuff inside is ever negative, get rid of the negative sign. If the stuff inside is positive, leave it that way.
x This is a variable, in fact, this is the quintessential variable. With this variable, and the function ƒ, you can build a graph.
a And this is a constant. More specifically, this is the constant that x approaches. I'll explain this cryptic comment later.
⇒ "Implies" This means that when the stuff to the left of the arrow is true, it automatically implies that the stuff to the right of the arrow is true.
L The letter L. But more specifically, this L is the answer to our limit problem. You'll see what I mean momentarily.
So, when you put all this together, what do you get?
For every epsilon greater than zero, there exists a delta, also greater than zero, such that when the absolute value of x minus a is less than delta, the value of the function evaluated at x, minus the limit, is less than epsilon.
All of this means nothing if you've never seen a limit before. This is what a limit looks like. I'd like to mention now that any terms you see in the below limit that happen to match the terms in the above delta epsilon proof, are one and the same.
lim ƒ(x) = L
I said earlier that the delta epsilon proof was also known as the precise definition of a limit, and it is. Instead of writing the above, you could write out that long line of gibberish, and the two statements would mean the same thing. Of course, no one expects you to do that, as "lim
" is much, MUCH easier to recognize, manipulate
, and understand
than that long string of mathematical symbols
. However, to fully understand what a limit is, you need to at least be aware of the delta epsilon thingy.
Let's step away from all the complicated math stuffs for a moment now, and just look at things in terms of graphs. We'll even use a specific example: ƒ(x) = 4x+3, x≠0
This defines a diagonal line (with a slope of 4) that would equal 3 when x is 0, except that we've expressly defined this function to have a hole there. This function has no value at zero. There is a break in the line.
The graph looks something vaguely resembling this:
Yeah, it's ugly, but it'll do for now. This graph is identical to that of 4x+3, except at zero. But you can get really really
close to 0 without reaching the point where the function refuses to respond. In fact, you can get infinitely close without ever touching zero. As close as you want. And as you get closer and closer to zero, the value of the function gets closer and closer to three.
Suppose you wanted to know just how close to zero you'd have to get so that ƒ(x) differs from three by less than zero point one (0.1)?
Well, we need some algebra here. We're looking for a maximum value of x, except we don't care if it's negative or positive, so we need the absolute value of x. We want the difference in the value of the function to be less that 0.1, and the slope is four, so our equation looks like...
|x| < (0.1)/4
= |x| < 0.025
This is starting to look like our definition, just a little bit, isn't it? Let me pull up that graph again, but this time without any numbers at all.
The distance between that hole there in the line and the question mark above or below it, that's ε. The distance between zero and the question marks to the left or right? That's δ.
When you look at it this way, the delta epsilon proof is defining a limit this way (in my own words):
A limit as x approaches a number equals L means that if you were to pick any number and evaluate the function at that number, I could pick a number even closer to the number x is approaching, and it would be closer to L than the number you picked.
That's it folks. That's the delta epsilon proof for you in a nutshell. Now, there is one more conditional
that I rarely see printed, and that's this: the initial epsilon you pick needs to be reasonably small
. I've tried to get my professors (and I've taken calculus twice, so I've had two chances at this) to precisely define "reasonably close" but both professors balked at the chore. So I will attempt to do it instead.
Reasonably close means that there can't be any critical points between the epsilon you pick and the target.
For most situations, you won't even need to think about this. Heck, you never actually need to use numbers in this proof, just the implication that there ARE potential numbers available should be satisfactory. But if you should ever want to plug numbers in, just for the heck of it, picking any number less than 1 is a safe bet in all but really confusing bloody complicated examples. So don't worry about it too much.
Brontosaurus informs me that the proof will still work for any epsilon, even those NOT reasonably close-- "if it holds for epsilon=1, then we have delta with |x-a|<δ⇒|ƒ(x)-L|<1 ⇒ |ƒ(x)-L|<1000, so it holds for ε=1000. But it is true that what one is really interested in is local behaviour, ie when epsilon is tending to 0." So take the above with a grain of salt. Still, both my professors thought it important enough to bring it up, take that as you will.
I can see a question forming in your brain there, assuming you are about to take calculus as a college or high school course...
"How will the professor test my knowledge of this proof?"
For those of you never planning to take a formal course in calculus, but still interested in the subject, your question might instead run along the lines of...
"How can I use this junk?"
I'm glad you asked. The answer to both questions is the same. You can use this to prove that a limit you calculated is accurate. You will never use this to actually calculate the limit itself, that would be just plain silly. Say you are told, perhaps on an exam, that...
ƒ(x) = x/5
lim ƒ(x) = 3/5
How would you prove that all this is true? Well, we need to show that for any epsilon, there is a corresponding
delta. The best way to show this is to show that there is a relationship
between epsilon and delta that always works. Here's how it goes:
|x-3|<δ ⇒ |ƒ(x)-(3/5)|<ε
Did you catch all that? All I did was write down the proof and substitute the actual numbers for a and L. I'm now going to go a step further and replace ƒ(x)
with the polynomial, and do a little reducing and algebraic manipulation.
|x-3|<δ ⇒ |(x/5) - (3/5)|<ε
|x-3|<δ ⇒ |(x-3)/5|<ε
|x-3|<δ ⇒ |x-3|<5ε
Notice any similarities between the delta side and the epsilon side? I'm sure you do, because you're smart. It's obvious here that δ=5ε, and the fact that this relationship exists proves the limit as true. And Bob's your uncle
. QED. That's the use of the delta epsilon proof. It shows that a limit is true