Limits: what a function heads towards

Approach a value from both sides, and see when a limit exists, and when it does not. About 9 minutes, A-level.

What you will do

1. Getting close without arriving. A limit asks what f(x) gets close to as x gets close to a number, even when f is undefined at that number.
2. When substitution works. If the function is smooth at the point, just put the number in.
3. 0 over 0. Sometimes substitution gives 00. That is a signal to look at nearby values, not an answer.
4. A famous limit. The table closes in on 1 from both sides, but no amount of table proves it: that needs a proper argument.
5. When the limit does not exist. If the two sides head to different places (or off to infinity) the limit does not exist.