Introduction to Limits Section 12.1 Introduction to Limits

Objective By following instructions students will be able to: Use the definition of a limit to estimate limits. Decide whether limits of functions exists. Use properties and operations of limits to find limits.

Mrs. Orca’s monthly checking account is represented by the graph Mrs. Orca’s monthly checking account is represented by the graph. The y-axis represents the amount of money in her account and the x-axis represents the day of the month. After 20 days, how much money will she have left in her account?

X Y 19.99 19.999 20 20.001 20.01 What is the purpose of a table? Challenges you to estimate a problem without a graph and teaches you how to visualize the limit if given a bigger domain, i.e. infinity.

Definition If f(x) becomes arbitrarily close to a unique number L as x approaches c from each side, the limit of f(x) as x approaches c is L. This is written as

EXAMPLE 1: Use a table to estimate numerically the limit.

EXAMPLE 2: Use a table to estimate numerically the limit.

EXAMPLE 3: Use a graphing calculator to estimate the limit.

EXAMPLE 4: Find the limit of f(x) as x approaches 3, where

U-TRY #1 Complete the table and use the result to numerically estimate the limit. Determine whether or not the limit can be reached. 1) 2) x 2.9 2.99 2.999 3 3.001 3.01 3.1 f(x) ? x 3.9 3.99 3.999 4 4.001 4.01 4.1 f(x) ?

Conditions Under Which Limits Do Not Exist The limit f(x) as xc does not exist if any of the following conditions is true. f(x) approaches a different number from the right side of c than from the left side of c. f(x) increases or decreases without bound as x approaches c. f(x) oscillates between two fixed values as x approaches c.

EXAMPLE 5: Show that the following limit does not exist.

EXAMPLE 6: Discuss the existence of the limit.

EXAMPLE 7: Discuss the existence of the limit.

Properties of Limits Let b and c be real numbers and let n be a positive integer. 1) 2) 3) 4) For n even and c>0.

Operations with Limits Let b and c be real numbers and let n be a positive integer, and let f and g be functions with the following limits. and 1)Scalar multiple: 2)Sum or difference: 3) Product:

Operations with Limits 4)Quotient: 5)Power:

EXAMPLE 8: Find each of the following limits. a) b) c) d)

U-TRY #2 Find the limit by direct substitution. a) b) 2) If and , find the sum, difference, product, and quotient of the limits.

Revisit Objective Did we… Use the definition of a limit to estimate limits? Decide whether limits of functions exists? Use properties and operations of limits to find limits?

Homework Pg 813 #s 3-49 EOO