Finding Limits Graphically and Numerically Lesson 2 2

Finding Limits Graphically and Numerically Lesson 2. 2

Average Velocity § Average velocity is the distance traveled divided by an elapsed time. A boy rolls down a hill on a skateboard. At time = 4 seconds, the boy has rolled 6 meters from the top of the hill. At time = 7 seconds, the boy has rolled to a distance of 30 meters. What is his average velocity?

Distance Traveled by an Object § Given distance s(t) = 16 t 2 § We seek the velocity • or the rate of change of distance 2 t § The average velocity between 2 and t

Average Velocity § Use calculator § Graph with window 0 < x < 5, 0 < y < 100 § Trace for x = 1, 3, 1. 5, 1. 9, 2. 1, and then x = 2 § What happened? This is the average velocity function

Limit of the Function § Try entering in the expression limit(y 1(x), x, 2) Expression variable to get close value to get close to § The function did not exist at x = 2 • but it approaches 64 as a limit

Limit of the Function § Note: we can approach a limit from • left … right …both sides § Function may or may not exist at that point § At a • right hand limit, no left • function not defined § At b • left handed limit, no right • function defined a b

Observing a Limit § Can be observed on a graph. View Demo

Observing a Limit § Can be observed on a graph.

Observing a Limit § Can be observed in a table § The limit is observed to be 64

Non Existent Limits § Limits may not exist at a specific point for § § a function Set Consider the function as it approaches x=0 Try the tables with start at – 0. 03, dt = 0. 01 What results do you note?

Non Existent Limits § Note that f(x) does NOT get closer to a particular value • it grows without bound § There is NO LIMIT § Try command on calculator

Non Existent Limits § f(x) grows without bound View Demo 3

Non Existent Limits View Demo 4

Formal Definition of a Limit § The • § For any ε (as close as you want to get to L) § There exists a (we can get as close as necessary to c ) View Geogebra demo

Formal Definition of a Limit § For any (as close as you want to get to L) § There exists a (we can get as close as necessary to c Such that …

Specified Epsilon, Required Delta

Finding the Required § Consider showing § |f(x) – L| = |2 x – 7 – 1| = |2 x – 8| < § We seek a such that when |x – 4| < |2 x – 8|< for any we choose § It can be seen that the we need is

Assignment § Lesson 2. 2 § Page 76 § Exercises: 1 – 35 odd