1. The Tangent Line Problem
What is the slope of the black line?
As the red point approaches the black point, the red secant line approaches the black tangent line, and
The slope of the secant line approaches the slope of the tangent line.

2. A Geometric Example of limit
Look at a polygon inscribed in a circle
As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.
n is the number of sides we can make some mathematical statements:
As n gets larger, then the red polygon gets closer to being a circle
As n approaches infinity, the polygon approaches the circle
The limit of the polygon , as n goes to infinity is the circle

3. Observing a Limit
Can be observed on a graph.
View Demo 3

4. Observing a Limit
Can be observed on a graph.

5. Non Existent Limits
f(x) grows without bound
View
Demo3

6. Non Existent Limits
View Demo 4

7. Formal Definition of a Limit
The
For any ε (as close as you want to get to L)
There exists a  (we can get as close as necessary to c ) •

8. Formal Definition of a Limit
For any  (as close as you want to get to L)
There exists a  (we can get as close as necessary to c Such that …

9. Specified Epsilon, Required Delta

10. Finding the Required  Consider showing
|f(x) – L| = |2x – 7 – 1| = |2x – 8| < 
We seek a  such that when |x – 4| < 
|2x – 8|<  for any  we choose
It can be seen that the  we need is

11. Example of limits at infinity
The function can diverge
The function doesn’t
converge to a single value but
its amplitude
keeps growing.
It diverges.

12. Examples of limits at x=0
The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

13. Try These
Do not use your calculator

15. Answers will be posted later.