As a key theorem, the Whitney Extension Theorem is essential for understanding smooth structures on manifolds with boundary.
Limits (Formal Definition)
Approaching ...
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)
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:
Let's work it out for x=1:
(12 − 1) (1 − 1) = (1 − 1) (1 − 1) = 0 0
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 |
| ... |
|
... |