Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer! We call that the limit of a function.

For example, what is the value of (x2-1)/(x-1) when x=1? (Notice in the graph there is actually a "hole" at that point).

Let us substitute "1" for "x" and see the result:

(12-1)/(1-1) = (1-1)/(1-1) = 0/0

But 0/0 is "indeterminate", meaning we can't determine its value, so instead of trying to work it out for x=1 let's try approaching it closer and closer:

x(x2-1)/(x-1)
0.51.50000
0.91.90000
0.991.99000
0.9991.99900
0.99991.99990
0.999991.99999
......

Now we can see that as x gets close to 1, then (x2-1)/(x-1) gets close to 2, so we say:

The limit of (x2-1)/(x-1) as x approaches 1 is 2

And it is written in symbols as:

We don't know the value at x=1 (it is indeterminate), but the limit has a value of 2

More Formal

But you can't just say a limit equals some value because it looked like it was going to. We need 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"

If 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"

 

Calculating "Close"

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" The solution is to use the absolute value.

"How Close" = |a-b|

Example 1: |4.01-4| = 0.01 
Example 2: |3.8-4| = 0.2 

And if |a-b| is small we know we are close, so we write:

"|f(x)-L| is small when |x-a| is small"

Needs Flash Player

And this animation shows you what happens with the function

f(x) = (x2 - 1) / (x-1)


  • as x approaches a=1,
  • f(x) approaches L=2

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:

"|f(x)-L|<when |x-a|<"

That actually says it! So if you understand that you understand limits ...

... but to be absolutely precise we need to add these conditions:

1)2)3)
it is true for any >0exists, and is >0x not equal to a means0<|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

From: To:
0<|x-a|<|f(x)-L|<

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, you can:

  1. Play around till you find a formula that might work
  2. 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


Start with:|(2x+4)-10|<
Simplify:|2x-6|<
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 ...

Start with:0<|x-3|<
Replace:0<|x-3|</2
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

DONE!

We have seen then that if we choose a 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