1. 2.1 Introduction to Limits
Motivating example
A rock falls from a high cliff.
The position of the rock is given by:
After 2 seconds:
average speed:
What is the instantaneous speed at 2 seconds?

2. We can use a calculator to evaluate this expression for smaller and smaller values of h.
We can see that the velocity approaches 64 ft/sec as h becomes very small. 1 80
0.1
65.6
.01
64.16
.001
64.016
.0001
.00001
We say that the velocity has a limiting value of 64 as h approaches zero.
(Note that h never actually becomes zero.)

3. Definition of Limit We write
and say “the limit of f(x) , as x approaches a, equals L”
if we can make the values of f(x) arbitrarily close to L
(as close to L as we like) by taking x to be sufficiently
close to a (on either side of a) but not equal to a.
In our example,

4. The limit of a function refers to the value that the function approaches, not the actual value (if any).
not 1

5. Left-hand and right-hand limits
We write
and say the left-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to to L by taking x to be sufficiently close to a and x less than a.
Similarly, we write
and say the right-hand limit of f(x) as x approaches a is equal to L if we can make the values of f(x) arbitrarily close to to L by taking x to be sufficiently close to a and x greater than a.

6. Note that
if and only if
and

7. Analyzing limits graphically
does not exist because the left and right hand limits do not match! 2 1 1 2 3 4
At x=1:
left hand limit
right hand limit
value of the function

8. Analyzing limits graphically
because the left and right hand limits match. 2 1 1 2 3 4
At x=2:
left hand limit
right hand limit
value of the function

9. Analyzing limits graphically
because the left and right hand limits match. 2 1 1 2 3 4
At x=3:
left hand limit
right hand limit
value of the function

10. Properties of Limits Suppose that c is a constant and the limits
and exist. Then

11. Properties of Limits (cont.)
Suppose that c is a constant, n is a positive integer and the limit
exists. Then

12. Direct Substitution Property
If f is a polynomial or a rational function and a is in the domain of f, then
Examples on the board.

13. Indeterminate forms Consider:
If we try to evaluate by direct substitution, we get:
Zero divided by zero can not be evaluated. The limit may or may not exist, and is called an 0/0 indeterminate form.
We can evaluate it by factoring and canceling: