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# 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

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.

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.