1. Algebraic Limits and Continuity
OBJECTIVE
Develop and use the Limit Properties to calculate limits.
Determine whether a function is
continuous at a point.

2. 1.2 Algebraic Limits and Continuity
LIMIT PROPERTIES:
If and and c is any constant, then we have the following limit properties:
L1.
The limit of a constant is the constant.

3. 1.2 Algebraic Limits and Continuity
LIMIT PROPERTIES (continued):
L2. The limit of a power is the power of that limit, and the limit of a root is the root of that limit.
where m is any integer, L ≠ 0 if m is negative,
where n ≥ 2 and L ≥ 0 when n is even.

4. 1.2 Algebraic Limits and Continuity
LIMIT PROPERTIES (continued):
L3. The limit of a sum or difference is the sum or difference of the limits.
L4. The limit of a product is the product of the limits.

5. 1.2 Algebraic Limits and Continuity
LIMIT PROPERTIES (concluded):
L5. The limit of a quotient is the quotient of the limits.
L6. The limit of a constant times a function is the constant times the limit of the function.

6. 1.2 Algebraic Limits and Continuity
Example 1: Use the limit properties to find
We know that
By Limit Property L4,

7. 1.2 Algebraic Limits and Continuity
Example 1 (concluded):
By Limit Property L6,
By Limit Property L1,
Thus, using Limit Property L3, we have

8. 1.2 Algebraic Limits and Continuity
THEOREM ON LIMITS OF
RATIONAL FUNCTIONS
For any rational function F, with a in the domain of F,

9. 1.2 Algebraic Limits and Continuity
Example 2: Find
The Theorem on Limits of Rational Functions and
Limit Property L2 tell us that we can substitute to find
the limit:

10. 1.2 Algebraic Limits and Continuity
Quick Check 1
Find the following limits and note the Limit Property you use at each step:
a.)
b.)
c.)

11. 1.2 Algebraic Limits and Continuity
Quick Check Solution a.)
We know that the
1.) Limit Property L4
2.) Limit Property L6
3.) Limit Property L4
4.) Limit Property L6
5.) Limit Property L1
6.) Combining steps 2.), 4.), and 5.) we get
Limit Property L3

12. 1.2 Algebraic Limits and Continuity
Quick Check 1 solution b.)
We know that
1.) Limit Properties L4 and L6
2.) Limit Property L6
3.) Limit Property L1
4.) Combine above steps: Limit Property L3
5.) Limit Property L6
6.) Limit Property L1
7.) Combine above steps: Limit Property L3
8.) Combine steps 4.) and 7.)
Limit Property L5

13. 1.2 Algebraic Limits and Continuity
Quick Check 1 solution c.)
We know that
1.) Limit Property L1
2.) Limit Properties L4 and L6
3.) Combine above steps: Limit Property L3
4.) Using step 3.)
Limit Property L2

14. 1.2 Algebraic Limits and Continuity
Example 3: Find
Note that the Theorem on Limits of Rational Functions
does not immediately apply because –3 is not in the
domain of
However, if we simplify first, the result can be
evaluated at x = –3

15. 1.2 Algebraic Limits and Continuity
Example 3 (concluded):
This means that the limits exist as x approaches -3, but the actual point does not (from previous slide).

16. 1.2 Algebraic Limits and Continuity
DEFINITION:
A function f is continuous at x = a if:
1) exists, (The output at a exists.)
2) exists, (The limit as exists.)
3) (The limit is the same as the output.)
A function is continuous over an interval I if it is
continuous at each point a in I. If f is not continuous at x = a, we say that f is discontinuous, or has discontinuity, at x = a.

17. 1.2 Algebraic Limits and Continuity
Example 4: Is the function f given by
continuous at x = 3? Why or why not? 1)
2) By the Theorem on Limits of Rational Functions,
3) Since f is continuous at x = 3.

18. 1.2 Algebraic Limits and Continuity
Example 5: Is the function g given by
continuous at x = –2? Why or why not? 1)
2) To find the limit, we look at left and right-side limits.

19. 1.2 Algebraic Limits and Continuity
Example 8 (concluded): 3)
Since we see that the
does not exist.
Therefore, g is not continuous at x = –2.

20. 1.2 Algebraic Limits and Continuity
Quick Check 2
Let Is g continuous at ? Why or why not?
1.)
2.) To find the limit, we look at both the left-hand and right-hand limits:
Left-hand:
Right-hand:
Since we see that does not exist.
Therefore g is not continuous at

21. 1.2 Algebraic Limits and Continuity
Quick Check 3a
Let Is h continuous at Why or why not?
In order for to continuous, So lets start by finding
So the However, , and thus Therefore is not continuous at

22. 1.2 Algebraic Limits and Continuity
Quick Check 3b
Let Determine c such that p is continuous at
In order for to be continuous at , So if we find , we can determine what c is. Let’s find :
So Therefore, in order for p to be continuous at ,

23. 1.2 Algebraic Limits and Continuity
Section Summary
For a rational function for which is in the domain, the limit as approaches can be found by direct evaluation of the function at
If direct evaluation leads to the indeterminate form , the limit may still exist: algebraic simplification and/or a table and graph are used to find the limit.
Informally, a function is continuous if its graph can be sketched without lifting the pencil off the paper.

24. 1.2 Algebraic Limits and Continuity
Section Summary Continued
Formally, a function is continuous at if:
The function value exists
The limit as x approaches a exists
The function value and the limit are equal
This can be summarized as
If any part of the continuity definition fails, then the function is discontinuous at