- 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

We now have enough tools to draw some complicated functions from scratch. Now we know how graphing calculators do it, and why they require the energy of four triple-A's.

When drawing a rational function *f*(*x*) from scratch, we need to know a lot of information, which can be nicely grouped into three big chunks.

- We need to know where
*f*has vertical asymptotes and/or holes. - We need to know the horizontal/slant/curvilinear asymptotes of
*f*, if any. - We need to know about values of
*f*. We found where*f*is undefined when we found the vertical asymptotes and holes; now we need to know where*f*(*x*) is 0, positive, and negative. We also want to know*f*(0), also called the*y*-intercept.

Example 1

Graph the function |

Example 2

Graph the function |

Example 3

Graph the function |

Example 4

Graph the function Label all asymptotes, intercepts, and holes. |

Exercise 1

Graph the function. Label all asymptotes, intercepts, and holes.

Exercise 2

Graph the function. Label all asymptotes, intercepts, and holes.

Exercise 3

Graph the function. Label all asymptotes, intercepts, and holes.

Exercise 4

Graph the function. Label all asymptotes, intercepts, and holes.

Exercise 5

Graph the function. Label all asymptotes, intercepts, and holes.