1. Limits Numerically 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. 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!!

2. Objectives To determine when a limit exists. To determine when a limit exists. To find limits using a graphing calculator and table of values. 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 intended height 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 from Calculus-Help.com When Does a Limit Exist? 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… 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. 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. 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 does not exist. If the height arrived at from the left does not match the height arrived at from the right, then the limit does not exist. Key Point: If a graph does not break at a given x -value, a limit exists there. 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.” 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. 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.” 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. 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. 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)  L as x  c

14. Finding Limits xg (x).9.99.999 xg (x) 1.1 1.01 1.001 6.71 6.9701 6.997 7.31 7.0301 7.003 = 7 If a function approaches the same value from both directions, then that value is the limit of the function at that point.

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

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

17. Conclusion A limit is the intended height of a function. 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 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 can exist if there is a hole in the graph. A limit will not exist if there is a break in the graph. A limit will not exist if there is a break in the graph.