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Limits (Evaluating)
Please read Limits (An Introduction) first
Quick Summary of Limits
Sometimes we can't work something out directly ... but we can see what it should be as we get closer and closer!
Example:
(x2 − 1)(x − 1)
Let's work it out for x=1:
(12 − 1)(1 − 1) = (1 − 1)(1 − 1) = 00
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:
Example Continued:
| 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 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
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