- Topics At a Glance
**Limits**- Functions Are Your Friend
**Graphing and Visualizing Limits**- Piecewise Functions and Limits
- One-Sided Limits
- Limits via Tables
- Limits via Algebra
- All About Asymptotes
- Vertical Asymptotes
- Finding Vertical Asymptotes
- Vertical Asymptotes vs. Holes
- Limits at Infinity
- Natural Numbers
- Limits Approaching Zero
- Estimating a Circle
- The Cantor Set and Fractals
- Limits of Functions at Infinity
- Horizontal, Slant, and Curvilinear Asymptotes
- Horizontal Asymptoes
- Slant Asymptotes
- Curvilinear Asymptotes
- Finding Horizontal/ Slant/ Curvilinear Asymptotes
- How to Draw Rational Functions from Scratch
- Comparing Functions
- Power Functions vs. Polynomials
- Polynomials vs. Logarithmic Functions
- Manipulating Limits
- The Basic Properties
- Multiplication by a Constant
- Adding and Subtracting Limits
- Multiplying and Dividing Limits
- Powers and Roots of Limits
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

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.

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.

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.

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 to 2 }*f*(*x*) = 5.

Let *f*(*x*) = 2 - 2*x*. Find the limit.

- lim
_{x to 2}*f*(*x*)

Exercise 9

Let *f*(*x*) = 2 - 2*x*. Find the limit.

- lim
_{x to 0 }*f*(*x*)

Exercise 10

Let *f*(*x*) = 2 - 2x. Find the limit.

- lim
_{x to 1 }*f*(*x*)