# At a Glance - Graphing and Visualizing Limits

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.

This is the basic idea behind limits. We look at what a function does as the independent variable, or input, gets closer and closer to some specified value.

#### Exercise 1

Let *y* = *f*(*x*) = sin(*x*). What is the limit of *f*(*x*) as *x* approaches 0?

#### Exercise 2

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

#### Exercise 3

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

#### Exercise 4

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

#### Exercise 5

Let *f*(*x*) = *x*^{2} -4*x* + 3. Graph *f*(*x*) and use the graph to find the limit of *f*(*x*) as *x* approaches 0.

#### Exercise 6

Let *f*(*x*) = *x*^{2} -4*x* + 3. Graph *f*(*x*) and use the graph to find the limit of *f*(*x*) as *x* approaches 3.

#### Exercise 7

Let *f*(*x*) =* x*^{2} -4*x* + 3. Graph *f*(*x*) and use the graph to find the limit of *f*(*x*) as *x* approaches 1.

#### Exercise 8

Although we say "the limit of *f*(*x*) as *x* approaches 2 is 5," we write

lim_{x → 2 }*f*(*x*) = 5.

Let *f*(*x*) = 2 – 2*x*. Find the limit, lim_{x → 2} *f*(*x*).

#### Exercise 9

Let *f*(*x*) = 2 – 2*x*. Find the limit, lim_{x → 0 }*f*(*x*).

#### Exercise 10

Let *f*(*x*) = 2 - 2*x*. Find the limit, lim_{x → 1 }*f*(*x*).