No.40

Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer!

Let's use this function as an example:

(x²-1)/(x-1)

And 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, 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)
0.5	1.50000
0.9	1.90000
0.99	1.99000
0.999	1.99900
0.9999	1.99990
0.99999	1.99999
...	...

Now we can 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 you get there, but as you get closer and closer the answer gets closer and closer to 2"

As a graph it looks like this: So, in truth, you cannot say what the value at x=1 is. But you can say that as you approach 1, the limit is 2.