## Dido525 3 years ago Find the values of a and b so that the following is true.

1. Dido525

I am thinking about L'Hospital's rule but I am not sure how I would apply it...

2. Rezz5

well, you can separate them, then solve them

3. Dido525

Yeah.... But I have two variables then.

4. Rezz5

are you sure it tends to infinity?

5. Dido525

Yep.

6. Dido525

0.

7. Dido525

oops.

8. Rezz5

why two variables?

9. Rezz5

if you differentiate the variable you get nothing from a, but b will still be present, you can get b right?

10. Dido525

$\lim_{x \rightarrow 0} (\frac{ \sin(2x) }{ x^3 }+a+\frac{ b }{ x^2 })=0$

11. Dido525

Yeah. I think you are right.

12. Rezz5

ahhh, so it tends to zero?

13. Dido525

Yep :P . I typed wrong.

14. Rezz5

no problem, you can separate them then do the limits, you can spot it by inspection, very easily

15. Dido525

Hmm Let me try...

16. Rezz5

have you done it?

17. Dido525

I can't solve the limit of: $\lim_{x \rightarrow 0}\frac{ \sin(2x) }{ x^3 }$

18. Rezz5

you can!! differentiate 3 times until the denominator is 1,

19. Rezz5

not 1, differentiate until the denominator is a constant

20. Rezz5

sin(2)=2sin(x)cos(x)

21. Dido525

I got -4/3 .

22. Rezz5

for?

23. Dido525

The limit.

24. Rezz5

sin(2x)/x^3 as x tends to infinity?

25. Rezz5

how ?

26. Dido525

To 0 :P .

27. Rezz5

lol!!! im losing it today...

28. Rezz5

but still, i would have thought it was zero

29. Dido525

Waitt.

30. Dido525

The Limit is infinity..... :( .

31. Rezz5

yep,

32. Dido525

We can't solve anything then.

33. Rezz5

we could, i mean if one of the variables was infinity

34. Dido525

But We can't do anything with infinity.... It's not a number.

35. Rezz5

wait are you sure it should tend to 0?

36. Dido525

Well I would assume it should tend to a finite value.

37. Rezz5

i suspect the limit should tend to infinity then for it so satisfy the limit

38. Rezz5

not necessarily, it can be infinity, let assume it was infinity, then we can do the question with ease. have you done analysis by any chance?

39. Dido525

No.....

40. Dido525

Here is the question.

41. Rezz5

okay, no problem, assume it is infinity because i think you were right the first time round, if it was infinity just plug in infinity to the differential you had computed

42. Dido525

If we did that then a would be infinity and b would not exist.

43. Rezz5

b would, it would be x^2,

44. Rezz5

one moment,

45. Dido525

but since b^2 tends to infinity the whole thing would be 0.

46. Dido525

x^2 sorry not b^2.

47. Rezz5

which would be 0, at when b/x^2 as x ->0

48. Dido525

Exactly. We can't work with that.

49. Rezz5

we could, if b = 0,

50. Rezz5

0/anynumber = 0

51. Dido525

But it's not. B/x^2 is zero, not b.

52. Dido525

We don't know what b is.

53. Rezz5

a could be -infinity, then this would satisfy the limit

54. Dido525

But infinity- infinity is not 0.

55. Rezz5

yes it is

56. Dido525

No it's not. Infinity is not a number. It's a concept.

57. Rezz5

that is true, i moment, i am trying to multi task, i jst remembered the proof that it isnt,

58. Dido525
59. Rezz5

okay, you need to apply l'hpitals rule over and over again, i would recon, but first you need to put all the equation under one common factor so $=\frac{\sin(2x) + ax^{3} + bx}{x^{3}}$

60. Rezz5

http://answers.yahoo.com/question/index?qid=20110406112720AARpKn1 here is a better and easier way of doing it the sint is taylor expansion i think, the rest is pretty self explanatory

61. Dido525

Lol. THanks :P .