1. Day 4

2. WARMUP—Please work the following upon arriving to class
WARMUP—Please work the following upon arriving to class. THANKS for getting right to work. 

3. ANSWERS

4. ANSWERS

5. HW Questions:

6. FROM earlier this week: KNOW THESE!!
What is different about these two?

7. Add this to your list of limits to KNOW.

8. A Sine Summary

9. A little trig practice

10. Answers--A little trig practice

11. Day 4 Infinite Limits Asymptotes
AP Calculus AB
Day 4
Infinite Limits
Asymptotes

12. An Infinite Limit-
Limit FAILS TO EXIST and the function increases without bound.
Limit FAILS TO EXIST and the function decreases without bound.

14. Reemphasizing the differences . . .

15. Determining positive or negative infinity without a calculator . . . . .
Test #s on either side of x-value and see which direction the y-values go. Pick #s that are close to the x-value.

16. Pick x values that are approaching 0 from the left.
Pick x values that are approaching 0 from the right. -1 -¾ -½ -¼ 1

17. Based on our work, determine the following limit.

18. You Try . . .

19. Based on our work, determine the following limit.

20. Calculus Definition: Vertical AsymptoteIf
then x=a is a vertical asymptote.

21. Example
Is there a vertical asymptote at x = 3? Justify your answer (with a CALCULUS justification).

22. Answer
There is a vertical asymptote at x=3 because

23. Tying Things Together . . .
Determine the following limit by showing the right hand and left hand limits using x-values.
Justify, using the calculus definition, that x=3 is a vertical asymptote.

24. End Behavior Models What happens as x approaches ∞ or -∞? Two Options:
Horizontal asymptotes
Increasing or decreasing without bound

25. Definition: Horizontal Asymptote

26. End Behavior Models
Comparing growth of numerator to the growth of denominator. The idea behind BOSTON.
1) The top is constant. The bottom is increasing without bound. The overall value is going to 0
Therefore, the limit is 0
2) The leading coefficients are the only factors in the slight difference in growth between the top and the bottom. So, the limit is the ratio of these coefficients.
3) The top is increasing without bound
The bottom is increasing at a slower rate. Therefore, the limit is increasing without bound
4) Same as the previous except approaching negative infinity
Therefore, the limit is decreasing without bound

27. Think of infinity as a race
Consider
BOSTON does not work!!!!  But, comparing growth does.
The top is increasing w/out bound, making the value larger.
The bottom is increasing w/out bound, making the value smaller.
Which is growing faster? Who wins the race? The top or the bottom?
ex grows faster than x, therefore, the limit is infinite.

28. Negative Exponents The top is now constant.
Consider
The top is now constant.
Sine is bounded but ex is increasing without bound making the value smaller.
Therefore, the limit is zero.

29. A CALCULUS justification. . . . NO BOSTON
Ex: What is the horizontal asymptote of ?
Answer:
The horizontal asymptote is y = 0 because

30. Practice: