12 1 Estimating Limits Graphically Day 1 OBJECTIVES

12. 1: Estimating Limits Graphically Day 1

OBJECTIVES Estimate limits of functions at a point. Estimate limits of functions at infinity. Essential Question How can I find the limits of a function numerically and graphically? 2

Warm-up #1: A Blast from the Past! State the end behavior by filling in the blanks.

Warm-up #2: A Blast from the Past! State the end behavior by filling in the blanks.

What is a limit? Informal Definition: If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from either side, the limit of f(x), as x approaches c, is L. 5

Limit The limit of f(x)… is L. Notation: as x approaches c… f(x) L x c

Calculating Limits There are three approaches to finding a limit: 1. Numerical Approach – Construct a table of values This Lesson 2. Graphical Approach – Draw a graph 3. Analytic Approach – Use Algebra or calculus Future Lesson

Example 1 -Numerically Complete the table to find the limit (if it exists). x 1. 999 2 2. 001 2. 01 f(x) 6. 859 7. 88 7. 988 8 8. 012 8. 12 2. 1 9. 261 If the function is continuous at the value of x, the limit is easy to calculate.

Example 1 -Graphically

Example 2 -Numerically Complete the table to find the limit (if it exists). Can’t divide by 0 x -1. 1 -1. 01 f(x) -2. 1 -2. 01 -1. 001 -2. 001 -1 DNE -. 999 -1. 999 -. 9 -1. 9 If the function is not continuous at the value of x, a graph and table can be very useful.

Example 2 -Graphically

Three Limits that Fail to Exist f(x) approaches a different number from the right side of c than it approaches from the left side.

Three Limits that Fail to Exist f(x) increases or decreases without bound as x approaches c.

Three Limits that Fail to Exist f(x) oscillates between two fixed values as x approaches c. Closest Closer Close x f(x) 0 -1 1 -1 DNE 1 -1 1

A Limit that DOES Exist If the domain is restricted (not infinite), the limit of f(x) exists as x approaches an endpoint of the domain.

Example 3 -Graphically Given the function t defined by the graph, find the limits at right.

Homework: 12. 1(day 1) Worksheet