1. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
An Introduction to Limits x y x y
Sketch the graph of the function.

2. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
An Introduction to Limits
Definition of a limit:
We say that the limit of f(x) is L as x approaches a and write this as
provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.

3. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
Example 1
Estimating a limit numerically
Estimate the value of the following limit. x y
Limits are asking what the function is doing around x = a, and are not concerned with what the function is actually doing at x = a.

4. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
Example 2
Finding a limit
Estimate the value of the following limit.

5. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
Example 3
Behavior that differs from the right and left
Estimate the value of the following limit.

6. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically 10 11 12 9 8 6 7 -5 -4 4 5 6 3 2 -3 1 5 3 -6 4 -7 -8 -9 -4 -5 1 -3 2 -2 -1
Example 4
Unbounded behavior
Estimate the value of the following limit.

7. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically 12 -5 -4 11 9 7 8 -3 10 -2 4 5 6 3 2 -1 1 6 5 4 -4 -6 -7 -9 -8 -3 -5 2 -2 1 3 -1
Example 5
Oscillating behavior
Estimate the value of the following limit. t
f (t)

8. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically
The Formal Definition of a Limit
Let f(x) be a function defined on an interval that contains x = a, except possibly at x = a. Then we say that,
if for every number e > 0 there is some number d > 0 such that
whenever

9. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically 12 -5 -4 11 10 7 8 9 -3 -2 4 5 6 3 2 1 -1 6 5 -4 -6 -7 -9 -8 -3 -5 3 -2 2 4 1 -1
Example 6
Finding a d for a given e
Given the limit
find d such that
whenever

10. Finding Limits Graphically and Numerically
1.2
Finding Limits Graphically and Numerically

11. 1.1
A Preview of Calculus
Pg. 46, 1.1 #1-11