1. Limits

2. Find and explain
Answer:
Does not exist
But why?!

3. Consider:
What happens as x approaches zero?
Numerically:
You can scroll down to see more values.

4. Graphically:

5. Looks like y=1
It appears that the limit of as x approaches zero is 1

6. …gives us this window
Turn the Axes off and then graph it again.

7. Looks like y=1
It again appears that the limit as x approaches zero is 1 even though it is not defined there.

8. Limit notation:
“The limit of f of x as x approaches c is L.”
So:

9. So what about this one:
But remember that:
…and so… 1 1 1

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

11. Properties of Limits:
Limits can be added, subtracted, multiplied, multiplied by a constant, divided, and raised to a power.
(See page 58 for details.)
For a limit to exist, the function must approach the same value from both sides.
One-sided limits approach from either the left or right side only.

12. 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

13. 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

14. 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

15. because at any endpoint, only a one-sided limit is necessary for the limit to exist.2 1 1 2 3 4
At x=4:
left hand limit
DNE
right hand limit
value of the function

16. Find
…with a hole at x = 2
And what does the graph look like?

17. The Sandwich Theorem: Show that: The maximum value of sine is 1, so
The minimum value of sine is -1, so
So:

18. By the sandwich theorem:
WINDOW

19. p