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) = 16t 2  We seek the velocity or the rate of change of distance  The average velocity between 2 and t 2 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(y1(x),x,2)  The function did not exist at x = 2 but it approaches 64 as a limit Expression variable to get close value to get close to

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

 Can be observed on a graph. Observing a Limit View Demo 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 grows without boundit grows without bound  There is NO LIMIT  Try command on calculator

Non Existent Limits  f(x) grows without bound View Demo3 View Demo3

Non Existent Limits View Demo 4 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 View Geogebra demo View Geogebra demo 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| = |2x – 7 – 1| = |2x – 8| <   We seek a  such that when |x – 4| <  |2x – 8|<  for any  we choose  It can be seen that the  we need is

Assignment  Lesson 2.2  Page 76  Exercises: 1 – 35 odd