Finding Limits Graphically and Numerically

1. Finding Limits Graphically and Numerically Lesson 2.2

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?

3. Distance Traveled by an Object • Given distance s(t) = 16t2 • We seek the velocity • or the rate of change of distance • The average velocity between 2 and t 2 t

4. 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

5. value to get close to variable to get close Expression Limit of the Function • Try entering in the expressionlimit(y1(x),x,2) • The function did not exist at x = 2 • but it approaches 64 as a limit

6. a b 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

7. Observing a Limit • Can be observed on a graph. ViewDemo

8. Observing a Limit • Can be observed on a graph.

9. Observing a Limit • Can be observed in a table • The limit is observed to be 64

10. 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?

11. 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 oncalculator

12. Non Existent Limits • f(x) grows without bound View Demo3

13. Non Existent Limits View Demo 4

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

15. 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 cSuch that …

16. Specified Epsilon, Required Delta

17. 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

18. Assignment • Lesson 2.2 • Page 76 • Exercises: 1 – 35 odd