Limits (Formal Definition)
Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!
(x2 − 1) (x − 1)
Let's work it out for x=1:
(12 − 1) (1 − 1) = (1 − 1) (1 − 1) = 0 0
Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is "indeterminate"), so we need another way of answering this.
So instead of trying to work it out for x=1 let's try approaching it closer and closer:
|x||(x2 − 1) (x − 1)|
Now we see that as x gets close to 1, then (x2−1) (x−1) gets close to 2
We are now faced with an interesting situation:
- When x=1 we don't know the answer (it is indeterminate)
- But we can see that it is going to be 2
We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"
The limit of (x2−1) (x−1) as x approaches 1 is 2
And it is written in symbols as:
So it is a special way of saying, "ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2"
As a graph it looks like this:
So, in truth, we cannot say what the value at x=1 is.
But we can say that as we approach 1, the limit is 2.
But instead of saying a limit equals some value because it looked like it was going to, we can have a more formal definition.
So let's start with the general idea.
From English to Mathematics
Let's say it in English first:
"f(x) gets close to some limit as x gets close to some value"
When we call the Limit "L", and the value that x gets close to "a" we can say
"f(x) gets close to L as x gets close to a"
Now, what is a mathematical way of saying "close" ... could we subtract one value from the other?
Example 1: 4.01 − 4 = 0.01 (that looks good)
Example 2: 3.8 − 4 = −0.2 (negatively close?)
So how do we deal with the negatives? We don't care about positive or negative, we just want to know how far ... which is the absolute value.
"How Close" = |a−b|
Example 1: |4.01−4| = 0.01
Example 2: |3.8−4| = 0.2
And when |a−b| is small we know we are close, so we write:
"|f(x)−L| is small when |x−a| is small"
And this animation shows what happens with the function
f(x) = (x2−1) (x−1)
f(x) approaches L=2 as x approaches a=1,
so |f(x)−2| is small when |x−1| is small.
Delta and Epsilon
But "small" is still English and not "Mathematical-ish".
Let's choose two values to be smaller than:
|δ||that |x−a| must be smaller than|
|ε||that |f(x)−L| must be smaller than|
Note: those two Greek letters (δ is "delta" and ε is "epsilon") are
so often used we get the phrase "delta-epsilon"
And we have:
|f(x)−L|<ε when |x−a|<δ
That actually says it! So if you understand that you understand limits ...
... but to be absolutely precise we need to add these conditions:
- it is true for any ε>0
- δ exists, and is >0
- x is not equal to a, meaning 0<|x−a|
And this is what we get:
For any ε>0, there is a δ>0 so that |f(x)−L|<ε when 0<|x−a|<δ
That is the formal definition. It actually looks pretty scary, doesn't it?
But in essence it says something simple:
f(x) gets close to L when x gets close to a
How to Use it in a Proof
To use this definition in a proof, we want to go
This usually means finding a formula for δ (in terms of ε) that works.
How do we find such a formula?
Guess and Test!
That's right, we can:
- Play around till we find a formula that might work
- Test to see if that formula does work
Example: Let's try to show that
Using the letters we talked about above:
- The value that x approaches, "a", is 3
- The Limit "L" is 10
So we want to know how we go from:
Step 1: Play around till you find a formula that might work
So we can now guess that δ=ε/2 might work
Step 2: Test to see if that formula works.
So, can we get from 0<|x−3|<δ to |(2x+4)−10|<ε ... ?
Let's see ...
Yes! We can go from 0<|x−3|<δ to |(2x+4)−10|<ε by choosing δ=ε/2
We have seen then that given ε we can find a δ, so it is true that:
For any ε, there is a δ so that |f(x)−L|<ε when 0<|x−a|<δ
And we have proved that
That was a fairly simple proof, but it hopefully explains the strange "there is a ..." wording, and it does show a good way of approaching these kind of proofs.