1. Limits An Introduction To: Changing Rates Secant and Tangent Lines &

2. Suppose you drive 200 miles, and it takes you 4 hours.
What is your average speed?
Your average speed is:
If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

3. A rock falls from a high cliff.
The change in position of the rock is given by:
What is your average speed after 2 seconds:
average speed:
What is the instantaneous speed at 2 seconds?

4. The tangent line problem
secant line
(secant means to cut)
(c, f(c))
(c, f(c)) is the point of tangency and
f(c ) – f(c) x
is a second point on the graph of f.

5. Tangent and secant lines

6. Limit of a Function Limit of a Function
Let f be a function and let a and L be real numbers L is the limit of f(x) as x approaches a, written
if the following conditions are met.
As x assumes values closer and closer (but not equal ) to a on both sides of a, the corresponding values of f(x) get closer and closer (and are perhaps equal) to L.
The value of f(x) can be made as close to L as desired by taking values of x arbitrarily close to a.

7. Limits We can find limits: graphically, numerically,
and algebraically.
The next slide finds the limit graphically.

8. Limit of a Function

9. Finding limits numerically

10. Limit of a Function--numerically
Values of may be computed near x = 2
x approaches 2 
 f(x) approaches 4 x
1.9
1.99
1.999
2.001
2.01
2.1
f(x)
3.9
3.99
3.999
4.001
4.01
4.1

11. Limit of a Function
The values of f(x) get closer and closer to 4 as x gets
closer and closer to 2.
We say that
“the limit of as x approaches 2 equals 4”
and write

12. Finding the Limit of a Polynomial Function (graphically)
Solution or from a graph of f(x).
We see that

13. Finding the Limit of a Polynomial Function
Example Find
Solution The behavior of near x = 1 can
be determined from a table of values,
x approaches 1  
f(x) approaches 2 x .9
.99
.999
1.001
1.01
1.1
f(x)
2.11
2.0101
2.001
1.999
1.9901
1.91

14. Finding the Limit of a Polynomial Function
Example Find where
Solution Create a graph and table.

15. Finding the Limit of a Polynomial Function
Solution
x approaches 3  
f(x) approaches 7
Therefore x
2.9
2.99
2.999
3.001
3.01
3.1
f(x)
6.8
6.98
6.998
7.004
7.04
7.4

16. Limits That Do Not Exist
If there is no single value that is approached
by f(x) as x approaches a, we say that f(x)
does not have a limit as x approaches a,
or does not exist.

17. Limit formal definition (Calculus B_C)