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## anonymous 4 years ago Find the values of a and b so that the following is true.

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1. anonymous

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

2. anonymous

well, you can separate them, then solve them

3. anonymous

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

4. anonymous

are you sure it tends to infinity?

5. anonymous

Yep.

6. anonymous

0.

7. anonymous

oops.

8. anonymous

why two variables?

9. anonymous

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

10. anonymous

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

11. anonymous

Yeah. I think you are right.

12. anonymous

ahhh, so it tends to zero?

13. anonymous

Yep :P . I typed wrong.

14. anonymous

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

15. anonymous

Hmm Let me try...

16. anonymous

have you done it?

17. anonymous

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

18. anonymous

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

19. anonymous

not 1, differentiate until the denominator is a constant

20. anonymous

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

21. anonymous

I got -4/3 .

22. anonymous

for?

23. anonymous

The limit.

24. anonymous

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

25. anonymous

how ?

26. anonymous

To 0 :P .

27. anonymous

lol!!! im losing it today...

28. anonymous

but still, i would have thought it was zero

29. anonymous

Waitt.

30. anonymous

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

31. anonymous

yep,

32. anonymous

We can't solve anything then.

33. anonymous

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

34. anonymous

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

35. anonymous

wait are you sure it should tend to 0?

36. anonymous

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

37. anonymous

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

38. anonymous

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. anonymous

No.....

40. anonymous

Here is the question.

41. anonymous

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. anonymous

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

43. anonymous

b would, it would be x^2,

44. anonymous

one moment,

45. anonymous

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

46. anonymous

x^2 sorry not b^2.

47. anonymous

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

48. anonymous

Exactly. We can't work with that.

49. anonymous

we could, if b = 0,

50. anonymous

0/anynumber = 0

51. anonymous

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

52. anonymous

We don't know what b is.

53. anonymous

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

54. anonymous

But infinity- infinity is not 0.

55. anonymous

yes it is

56. anonymous

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

57. anonymous

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

58. anonymous
59. anonymous

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. anonymous

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. anonymous

Lol. THanks :P .

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