1. 14B Limits Involving Infinity

2. We need to think about what happens to a function not at a certain value, but at extremes like infinity.

3. Complete the following table for the function: x f(x) 2½ 10 100 200 500 700 1000 10,000 100,000

4. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

5. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

6. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

7. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

8. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

9. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

10. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

11. Complete the following table for the function: x f(x) 2½ 100.10 1000.01 2000.005 5000.002 7000.00143 10000.001 10,0000.0001 100,0000.00001

12. What about if it goes towards negative infinity?

13. Let’s look at it graphically

14. If the degree of the denominator is greater than the numerator, then. If the degree is Bigger On Bottom its 0. BOB0. (y = 0)

15. We can ignore the (-9) and the (+ 12). They really do not add anything to the graph when you go to ±∞

16. So, let’s look at this graphically

17. As the graph approaches ±∞, what is the “height” of the graph?

19. If the degree of the denominator is greater than the numerator, then. If the degree is Bigger On Bottom its 0. BOB0. (y = 0)

20. Bigger On Top, there’s No horizontal asymptote. BOTN

24. Degrees of both the numerator and denominator are equal Then divide the leading coefficients. That’s your horizontal asymptote. EATS-D/C.

27. Page 349 1a) i. There are going to be some new symbols. As x  0 - f(x)  -∞ Vertical Asymptote x = 0 As x  0 + f(x)  ∞

28. Page 349 1a) i. x  ∞ f(x)  0 + Vertical Asymptote y = 0 x  -∞ f(x)  0 -

29. Page 349 1a) ii.

30. Homework Page 349 Numbers 1 - 3

31. Chapter 14 C “ish”

32. Problem You are hanging out at your girlfriend’s place and she goes to get you a couple of slices of pizza. Her annoying cat is looking at you and you get this great idea for a calculus experiment. Because cats have nine lives and always land on their feet, you figure no harm can come from this. So, you drop her cat out of her second-story room window. Here’s the formula that tells you how far the cat “jumped” after a given number of seconds (ignoring air- resistance):

33. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5

34. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2

35. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32

36. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48

37. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48 64

38. Distance = (rate)(time) Please make a table of the average speed for the first 5 seconds. tAvg. Speed 00 1 2 3 4 5 16 h(t) = 16t 2 32 48 64 80

39. What if you wanted to determine the cat’s speed exactly 2 seconds after it “jumped”?

40. The cat is traveling about 16 fps. This is nice, but what if I wanted to know the EXACT speed 1 second after the jumped. The table gives an average. I want to know EXACT speed. See, the cat speeds up between 1 and 2 seconds and so on. Let’s look at the speed between 1.5 and 1 second

41. The cat is traveling about 40 fps.

42. The cat is traveling about 38.4 fps.

43. The cat is traveling about 27 fps.

44. The cat is traveling about 35 fps.

45. The cat is traveling about 32.0 fps.

46. As t gets closer and closer to 1 second, the average speed appears to get closer and closer to 32 fps.

48. The homework is from Chapter 14 B And The Handout