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
Example 2: 3.8 − 4 = −0.2
Hmmm ... negatively close? That doesn't work ... we really need to say "I don't care about positive or negative, I 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:
|that |x−a| must be smaller than|
|that |f(x)−L| must be smaller than|
(Note: Those two greek letters, δ is "delta" and ε is "epsilon", are often
used for this, leading to the phrase "delta-epsilon")
And we have:
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 >0||exists, and is >0||x not equal to a means 0<|x−a||
And this is what we get:
"for any>0, there is a >0 so that |f(x)−L|<when 0<|x−a|<"
That is the formal definition. It actually looks pretty scary, doesn't it!
But in essence it still says something simple: when x gets close to a then f(x) gets close to L.
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 (in terms of ) 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 works.
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 do we go from: ||0<|x−3|<||to|||(2x+4)−10|<|
Step 1: Play around till you find a formula that might work
|Move 2 outside:||2|x−3|<|
|Move 2 across:|||x−3|</2|
So we can now guess that =/2 might work
Step 2: Test to see if that formula works.
So, can we get from 0<|x−3|< to |(2x+4)−10|< ... ?
Let's see ...
|Move 2 across:||0<2|x−3|<|
|Move 2 inside:||0<|2x−6|<|
|Replace "−6" with "+4−10"||0<|(2x+4)−10|<|
Yes! We can go from 0<|x−3|< to |(2x+4)−10|<by choosing =/2
We have seen then that given we can find a , so it is true that:
"for any, there is a so that |f(x)−L|<when 0<|x−a|<"
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 you a good way of approaching these kind of proofs.