Functions, Graphs, and Limits
The Basic Properties
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
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.
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
As x gets closer to 3, f(x) gets closer to 3 since x and f(x) are the same thing.
The picture of f(x) = x is a line:
From the picture we can see that as x gets closer to 3, f(x) also gets closer to 3.
We conclude that = 3.