1. 1.5: Limits We will learn about: limits, finding limits,
one-sided limits, and
unbound behavior of functions

2. Limits
The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of one’s endurance” or the “limit of one’s patience.” In mathematics, the word “limit” has a similar but more precise meaning.

3. Limits
Given a function f(x), if x approaching 3 causes the function to take values approaching (or equaling) some particular number, such as 10,
then we will call 10 the limit of the function
and write

4. How to find limits? Graphically – look at the picture/graph
Numerically – analyze the table of values
Analytically – algebraic techniques

5. 1. Graphically Find the limit:
Draw graph: What is happening to y as x gets closer to 1?

6. 2. Numerically
This table shows what f (x) is doing as x approaches 3. Or, we have the limit of the function as x approaches 3. We write this procedure with the following notation.
We write: 10
General notation:
Or, as x → c, then f (x) → L
If the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c (on either side of c). x 2
2.9
2.99
2.999 3
3.001
3.01
3.1 4
f (x) 8
9.8
9.98
9.998 ?
10.002
10.02
10.2 12

7. Practice:
Find the limit:

8. Things to Note:
Looking at what happens to f (x) when x approaches c.
It doesn’t matter what happens AT c, just what happens as x gets closer to c.
x must approach c from either side of c If the value when x approaches c from the left does not equal the value when x approaches c from the right, then…
The Limit Does Not Exist!!!!

9. Definition
If f (x) becomes arbitrarily close to a single number L as x approaches c from either side, then
This is read as
“the limit of f (x) as x approaches c is L.”

10. Properties of Limits
Let b and c be real numbers, and let n be a positive integer.

11. Operations with Limits
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

12. Practice – Evaluate Analytically
= -85
= -2
= 4

13. What happens now? is an indeterminate form.
Use the Replacement Theorem:
Let c be a real number and let f (x) = g (x) for all If the limit of g (x) exists as x approaches c, then the limit of f (x) also exists and

14. Factor and Cancel
Practice – Pg. 93 # 42, 44, 46, 48

15. Rationalize the Numerator
Practice – Pg. 91 # 5, 6 –
algebraically

16. Practice:

17. One-Sided Limits While evaluating limits from graphs or tables:
We saw: the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3–, and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3+.
Such limits are called one-sided limits. x 2
2.9
2.99
2.999 3
3.001
3.01
3.1 4
f (x) 8
9.8
9.98
9.998 ?
10.002
10.02
10.2 12

18. One-Sided Limits
Limit from the Left
Limit from the Right

19. Practice! Evaluate the following
= -1
= 1
Practice – Pg. 93 # 52, 54

20. Existence of a Limit
If f (x) is a function and c and L are real numbers, then
if and only if
both the left and right limits exist
and are equal to L

21. Unbounded Behavior
Another way for the limit to fail to exist:

22. Function is unbounded – Limit does not exist
Practice – Pg. 93 # 62, 64