Limits Numerically

1. Limits Numerically The height of the line y=2 is always 2, so the “intended height” or “where it is heading towards” is always going to be 2!! Warm-Up: What do you think the following limit equals? If you are unsure at least recall what a limit is and see if that helps direct you.

2. Objectives • To determine when a limit exists. • To find limits using a graphing calculator and table of values. TS: Explicitly assessing information and drawing conclusions.

3. What is a limit? A limit is the intendedheight of a function.

4. How do you determine a function’s height? Plug an x-value into the function to see how high it will be.

5. Can a limit exist if there is a hole in the graph of a function? Yes, a limit can exist if the ultimate destination is a hole in the graph.

6. Limit Notation The limit, as x approaches 2, of f (x) is 4. or The limit of f (x),as x approaches 2, is 4.

7. Video Clip fromCalculus-Help.com When Does a Limit Exist?

8. When does a limit exist? • A limit exists if you travel along a function from the left side and from the right side toward some specific value of x, and… • As long as that function meets in the middle, as long as the heights from the left AND the right are the same, then the limit exists.

9. When does a limit not exist? • A limit will not exist if there is a break in the graph of a function. • If the height arrived at from the left does not match the height arrived at from the right, then the limit doesnotexist. • Key Point: If a graph does not break at a given x-value, a limit exists there.

10. One Sided Limits

11. Right-hand Limit:the height arrived at from the right • Read as: “The limit of f (x) as x approaches 4 from the right equals 2.” • This means x approaches 4 with values greater than 4.

12. Left-hand Limit: the height arrived at from the left • Read as: “The limit of f (x) as x approaches 4 from the left equals 1.” • This means x approaches 4 with values less than 4.

13. General Limit • A general limit exists on f (x) when x = c, if the left- and right-hand limits are both equal there. Mathematic Notation: In other words: f (x)  Las x  c

14. Finding Limits = 7 = 7 = 7 If a function approaches the same value from both directions, then that value is the limit of the function at that point. x g (x) x g (x) .9 6.71 1.1 7.31 .99 6.9701 1.01 7.0301 .999 6.997 1.001 7.003

15. Finding Limits = DNE or NL = 3 = –3 If the Left-hand limit and the Right-hand limit are not equal, the general limit does not exist. x h (x) x h (x) –1.1 3.1 –.9 –2.9 –1.01 3.01 –.99 –2.99 3.001 –1.001 –.999 –2.999

16. Finding Limits = DNE or NL = NL = NL x j (x) x j (x) If either the Left-hand limit, Right-hand limit, or both do not exist, the general limit will not exist. 2.9 –44.1 3.1 56.1 2.99 –494 3.01 506.01 2.999 –4994 3.001 5006

17. Conclusion • A limit is the intended height of a function. • A limit will exist only when the left- and right-hand limits are equal. • A limit can exist if there is a hole in the graph. • A limit will not exist if there is a break in the graph.