Exploring Limits and Behaviors in Math

1. Problem of the Day Find the distance and the 
midpoint between the 
points (2, 1) and (4, 5).

2. Problem of the Day Find the distance and the 
midpoints between the 
points (2, 1) and (4, 5). 2 2 (4 - 2) + (5 - 1) = 2 5 Distance = ( ) 4 + 2 5 + 1 = (3, 3) Midpoint = , 2 2

3. An Introduction to Limits (Graphically) 3 f(x) = x - 1 (x = 1) x-1 As x approaches 1 from the left, f(x) 
approaches 3. As x approaches 1 from the right, f(x) 
approaches 3.

4. An Introduction to Limits (Graphically) (x = 1) 3 f(x) = x - 1 x-1 You can conclude that lim f(x) = 3 x 1 General notation lim f(x) = L x c

5. An Introduction to Limits (Numerically) 3 f(x) = x - 1 (x = 1) x-1 You can conclude that lim f(x) = 3 x 1

6. An Introduction to Limits (Algebraically) f(x) = x + 4, find the limit when x approaches 2 f(2) = 2 + 4 = 6 You can conclude that lim f(x) = 6 x 2

7. An Introduction to Limits (Limits that Fail to Exist) What if the behavior as you approach 
from the left and right is different? Different Behavior

8. An Introduction to Limits (Limits that Fail to Exist) What about unbounded behavior? lim = 1 2 x 2 (x - 2)

9. An Introduction to Limits (Limits that Fail to Exist) What about unbounded behavior? Will the limit ever approach a real number? lim = 1 2 x 2 (x - 2)

10. An Introduction to Limits (Limits that Fail to Exist) What about oscilating behavior? lim = cos 1 ) ( x 2 x .03

11. An Introduction to Limits (Limits that Fail to Exist) What about oscilating behavior? The values jump from positive to negative. lim = cos 1 ) ( x 2 x 0

12. An Introduction to Limits Limits that fail to exist - 1) behavior that approaches  different numbers from the  right 
and left 2) unbounded behavior 3) oscilating behavior

13. An Introduction to Limits http://archives.math.utk.edu/visual.calculus/1/limits.16/tut1-flash.html

17. Example

19. Practice Epsilon delta definition practice practice with a general epsilon