# Functions, Graphs, and Limits Topics

## Functions, Graphs, and Limits explanations, examples, practice problems. Ready? Let’s do this.

#### Limits

Functions are usually written something like this: y = f(x).x is called the independent variable, while y is called the dependent variable. The indepe...

#### All About Asymptotes

Some things are untouchable. The Mona Lisa, MC Hammer...asymptotes? An asymptote is a line to which a function gets infinitely close. There are several different types of asymptotes, and it will...

#### Limits at Infinity

Newsflash: when taking limits we don't have let x approach some fixed number. We can also see what happens as x blows up to ±∞.#mindblownWe write these limits asThese limits look at where the fu...

#### Limits of Functions at Infinity

When we find the limit of a function f(x) as x goes to infinity, we're answering the question "What value is f(x) approaching as x gets bigger and bigger and bigger...?''A rockin' example of this i...

#### Horizontal, Slant, and Curvilinear Asymptotes

We've talked about vertical asymptotes where y runs off forever, but whoever said x can't ride off into the sunset (or the negative sunset), too? In this section we'll talk about other types of a...

#### How to Draw Rational Functions from Scratch

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 fu...

#### Comparing Functions

Power the race horse, Polly the parrot, and Log the log walk...or roll...into a bar. After a couple root beers, Log challenges the others to a race on a horse track. Who will we bet on? Power, of...

#### Manipulating Limits

Why can't limits all get along? Ignoring them is not the answer. Sometimes we need to show them who's boss by separating them, calming them down, and then bringing them back together again.There...

#### In the Real World

Sometimes measurements and results aren't perfect. This is true in science, engineering, and life. How close to 16.9 fluid ounces of Pythagor-ade can a plastic bottle hold in order to allow 16.9...