1. Limits Graphically & Limit Properties
Day 1
Limits Graphically & Limit Properties

2. What is a limit?
A limit describes how the output values of a function behave as input values approach some given #, “c”
Notation:
Read as “limit of f(x) as x approaches c is equal to L”

3. 3 Kinds of limits THE Limit (double-sided limit) Left-hand limit
Limit of f(x) as x approaches c from either direction.
Only exists if left-hand and right-hand limits are the same.
Left-hand limit
Limit of f(x) as x approaches c from the left side.
Right-hand limit
Limit of f(x) as x approaches c from the right side.

5. Common Misconception #1
A function does not have to be defined at “c” in order for the limit to exist.

6. Common Misconception #2
If a function is defined at “c”, f(c) does not necessarily have to equal L.

7. Practice

8. Practice

9. Practice

10. Practice 10

11. Draw a graph such that

12. Draw a graph such that

13. Draw a graph such that

14. Draw a graph such that

15. Draw a graph such that

16. Draw a graph such that

17. Two Cases for When the Limit is D.N.E. (Does Not Exist)
Behavior differs from the left and right
Oscillating Behavior
Ex/

18. Increase or decrease without bound
Unbounded Behavior
Increase or decrease without bound
Can you think of a parent function that would fall in this category?
More descriptive than DNE

19. Some Basic Limits
The limit of a constant function is the constant.

20. The limit at any x-value on the line y=x IS the x-value itself.

21. One more:
The limit at any x-value of a function f(x)=xn (where n is an integer) is the x-value raised to the nth power.

22. Practice:

23. Theorem: Properties of limits

24. Theorem: Properties of limit
The limit of a sum is equal to the sum of the limits
The limit of a difference is equal to the difference of the limits

25. Theorem: Properties of limit
The limit of a product is equal to the product of the limits

26. Theorem: Properties of limit
The limit of a quotient is equal to the quotient of the limits

27. Theorem: Properties of limit