1. We will find limits algebraically
1.2 Computing Limits
We will find limits algebraically

2. Theorem 1.2.1
Let a and k be real numbers.

3. Theorem 1.2.1
Let a and k be real numbers.

4. Theorem 1.2.1
Let a and k be real numbers.

5. Theorem 1.2.1
Let a and k be real numbers.

6. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

7. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

8. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
The limit of a sum is the sum of the limits.

9. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

10. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
The limit of a difference is the difference of the limits.

11. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

12. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
The limit of a product is the product of the limits

13. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

14. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
The limit of a quotient is the quotient of the limits

15. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:

16. Theorem 1.2.2 Let a be a real number, and suppose that
That is, the limits exist and have values L1 and L2, respectively. Then:
The limit of an nth root is the nth root of the limit.

17. Constants
A constant factor can be moved through a limit symbol.

18. Example 5
Find

19. Example 5
Find

20. Example 5
Find

21. Theorem 1.2.3
For any polynomial
and any real number a,

22. Rational Functions
We will have three situations when dealing with rational functions.
The denominator is not zero.

23. Rational Functions
We will have three situations when dealing with rational functions.
The denominator is not zero.
The denominator and the numerator are both zero.

24. Rational Functions
We will have three situations when dealing with rational functions.
The denominator is not zero.
The denominator and the numerator are both zero.
The denominator is zero and the numerator is not.

25. Denominator not Zero
When the denominator is not zero, we evaluate the numerator and denominator and for this problem use theorem 1.2.2d: the limit of a quotient is the quotient of the limit.

26. Denominator not Zero
When the denominator is not zero, we evaluate the numerator and denominator and for this problem use theorem 1.2.2d: the limit of a quotient is the quotient of the limit.

27. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph. Even though the value doesn’t exist, the limit does.

28. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph.
Factor, cancel, evaluate.

29. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph.
Factor, cancel, evaluate.

30. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph.
Factor, cancel, evaluate.

31. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph.
Factor, cancel, evaluate.

32. Numerator and Denominator both Zero.
We know that when the numerator and denominator are both zero, there is a hole in the graph.
Rationalize the denominator, cancel, evaluate.

33. Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.

34. Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.
Sign analysis- Determine what the graph is doing around the asymptote.

35. Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.
Sign analysis
___+___|_____-_____|___+___|___-___

36. Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.
Sign analysis
___+___|_____-_____|___+___|___-___

37. Denominator is Zero, the Numerator is not
We know that when the denominator is zero and the numerator is not, there is a vertical asymptote.
Sign analysis
Remember, all three answers mean that the limit does not exist. The first two answers tell us exactly what the function is doing and why the limit DNE.

38. Limits of Piecewise Functions
Let
Find

39. Limits of Piecewise Functions
Let
Find

40. Limits of Piecewise Functions
Let
Find

41. Limits of Piecewise Functions
Let
Find

42. Homework
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