Study Guide

Functions, Graphs, and Limits

Functions, Graphs, and Limits Introduction

Imagine our friend Moe has been dared, no, double dared. No. Triple Dog Dared. He must ride the dreaded spinning teacups in Disney World and then try to walk west toward Cinderella's Castle. Did we mention Moe can barely walk straight on a normal day?

Poor Moe survives the ride, but things aren't looking good. He's looking a little green, but determined. We guide him to safety, then he's off. Moe starts in a generally southward direction but soon realizes the castle is out of sight. He changes course, over-corrects, and heads generally northward. He changes course again, over corrects, but is a little closer to being castle-bound by continuing on a southwest path. Imagine that this goes on for a really long time. Moe is getting closer and closer to walking completely west toward the castle. He may never walk perfectly west, but he'll at least get close.

Welcome to the wild world of limits. A world where we try to approach stuff—sometimes with a hit, sometimes with a miss.

Functions, Graphs, and Limits Resources

Websites

How to Solve Limits with a Calculator: For Dummies
Ever read German For Dummies or Stock Investing For Dummies? Turns out they made Limits For Dummies, too. With this website, learn about limits and follow one quick link to learn how to perfect your soufflé—and other than calculus, everyone wants to know how to make a great soufflé.

How to Solve Limits at Infinity by Using Horizontal Asymptotes
Also from For Dummies, this link shows you how to relate horizontal asymptotes to limits of functions. It's sort of like relating long lines to the seemingly endless wait time for the most popular roller coaster at the amusement park, but with a slightly less exciting payoff.

Basic Limit Laws
Here's a quick guide to manipulating limits. If only your math teacher could have made geometric proofs this easy.

Wolfram MathWorld: Division by Zero
Yes, everyone's been lying to you. Sometimes, you actually can divide by zero.

(x, why?) 373: Speed Limit
Imagine driving down the road and seeing this. If you don't solve this fast enough, you better hope the officer failed the limits exam in his calculus class.

Videos

Evaluating Limits Problems 1: Based on Algebra of Limits of Calculus
Here's some extra explanation on how to solve a few limit problems. Now you know we weren't just waving our hands like a poorly performed magic trick.

Limits at Infinity: Basic Idea and Shortcuts!
The paradox of a shortcut to the limit at infinity is complicated enough to spark a drawn-out debate with your philosophy professor. Alternatively, you could watch this video and learn a few neat tricks to determine limits in special cases.

Finding Limits of Trigonometric Functions in Infinity: Infinity Form & When x Tends to 90 Degrees
Just when you thought you were through with trigonometry, it rears its ugly head. Don't worry—we have your back. This video will teach you the basics of trig function limits.

Finding Limits Using the TI-89
A stranger walks up to you and asks you to solve a difficult limit problem or hand over your lunch money. (True story. Mostly.) If you don't have a laptop handy, this video will show you how you can nerd it out and use a calculator instead.

Games and Tools

Online Limits Calculator
Have you ever gone down a road without knowing which way it leads or if it even ever ends? (So that's why nobody wants to ride with us when we volunteer to drive.) Some limits of functions can feel that way, too. This limit-finding tool is the GPS you need to find your way to any limit.