# Functions, Graphs, and Limits

### Topics

It's super helpful to plug numbers into the function and see the output. It's even more helpful to graph the results. Try to draw or imagine how a function actually looks. Is it really hip to be *x*^{2}? Draw the graph and decide for yourself.

### Sample Question

Let *y* = *f*(*x*) = *x*^{3 }- 2. As *x* gets close to zero, what does *y* approach?

As *x* approaches zero, *y* approaches *-2*. There are several different ways to say this:

- As
*x*gets close to 0,*y*gets close to*-2*. - As
*x*gets close to 0,*f*(*x*) gets close to*-2*.

- As
*x*approaches 0,*y*approaches*-2*. - As
*x*approaches 0,*f*(*x*) approaches*-2*.

- As
*x*goes to 0,*y*goes to*-2*. - As
*x*goes to 0,*f*(*x*) goes to*-2*.

Each of these phrases mean the same thing. Here's yet another way to say it:

The **limit** of *f*(*x*) as *x* approaches 0 is *-2*.

We know what "*x* approaches 0" means. The ** limit** of *f*(*x*) is the value *f*(*x*) is getting close to.

We can have *x* approach other numbers besides 0.

### Sample Problem

Let *y* = *f*(*x*) = cos(*x*). What is the limit of *f*(*x*) as *x* approaches 2π?

Moving *x* around, we see that as *x* gets closer to 2π, *f*(*x*) gets close to 1. The limit of *f*(*x*) as *x* approaches 2π is 1.