Warm Up

Getting Started Use your calculator to graph the following equation: What do you notice about the graph? Look closely!!!! Change your window to xmin = 1 and xmax = 3 xscl =.25 and see what happens!

Chapter 11 Limits and An Introduction to Calculus

Introduction to Limits If f(x) becomes arbitrarily close to a unique number L as x gets closer to c, then the limit of f(x) as x approaches c is L. This is denoted by

Examples Use a table to evaluate the following limits: .. Also, note that f(4) = 3 .. If we factor the numerator we can easily see that the limit = 4 Also, note that f is undefined at 2.

Using Graphs to Evaluate Limits The limit is –1. Also note that cos  = -1

Limits That Fail to Exist Since f(x) = 1 if x > 1 and f(x) = -1 if x < 1, the limit fails to exist. Since f(x)   if x > 0 and f(x)  -  if x < 0, the limit fails to exist.

Operations with Limits

Properties of Limits and Direct Substitution

Examples

More Examples 1.Evaluate the limit: 2.Evaluate the limit:

We need to make a single fraction as our numerator Now bring back our denominator

Warm Up Find the limit, if it exists, both algebraically and graphically:

At First Glance… As we saw in 11.1, sometimes, it would appear as though the limit does not exist. However, after simplifying, it actually does exist. We are going to examine different ways that we can simplify and then find the limit actually does exist! In other words, we are going to do some ALEGBRA!

Dividing Out Technique When direct substitution yields 0/0, we need to divide out (simplify) the fraction first. Example:

Rationalizing Technique Evaluate the following limit:

One-Sided Limits Right and Left hand limits… The if and only if

Example: Find the limit of f(x) as x approaches 2:  From the graph we see that the limit from the right is 1 and the limit from the left is –1. Therefore the limit of f(x) as x approaches 2 fails to exist.