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Functions, Graphs, and Limits

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.

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.

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.

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