Basic Property 1
If c is a real number, then
Think of this as taking the limit of the constant function f(x) = c. No matter what we plug in for x, we get c as the output. If we made a table
| x | f(x) |
|---|---|
| x1 | c |
| x2 | c |
every number in the f(x) column would be c. As x gets closer to a, f(x) gets closer to (or stays equal to) c.
As long as a is a real number, it doesn't even matter what a is; c is all that matters.
Sample Problem
Basic Property 2
Using actual numbers, what is 
?
We're looking at the limit as x approaches 3 of the function f(x) = x. If we make a table, we find
| x | f(x) |
|---|---|
| 2.99 | 2.99 |
| 2.999 | 2.999 |
| 3.0001 | 3.0001 |
| 3.001 | 3.001 |
| 3.01 | 3.01 |
As x gets closer to 3, f(x) gets closer to 3 since x and f(x) are the same thing, So we can conclude that = 3.
There's plenty more properties to come. Stay tuned.