Aug 19 - Limits introduction

We talked about limits today in a very informal way. The main idea that you need to remember is that the statement

lim f(x) = L
x->c

means that "for x values that are close to c, the y values are close to L."

What "close to" means is not well defined here. There is analysis that we will talk about that makes more specific the intuitive idea of "close to," but really for our purposes in calculus, the intuition should be fine.

Given the graph of y = f(x) here (click for an enlargement) :


Evaluate the limit of f(x) for each of the following:
x->0, the limit is 1
x->-3, the limit is 3
x->1, the limit is 2. this limit problem is somewhat bothersome b/c of the presence of the point at (1, 1). understand that the limit only refers to points "near" x = 1, not necessarily "at" x = 1.
x->-1, the limit does not exist due to different left/right behavior

Note that if you were asked for f(1), you could say f(1) = 1. If you were asked for f(-3), you would say that f(-3) is undefined.

Recall that there are three reasons for which a limit does not exist

• different left/right behavior
• unbounded behavior
• oscillating behavior

We will investigate these more tomorrow.