## anonymous one year ago Another limit question...

1. anonymous

$\lim_{n \rightarrow \infty}\ln\left( 1+ \frac{ 4-\sin(x) }{ n } \right)^n$

2. anonymous

My approach was that I raised it to the power of e and then I Just found the limit of the stuff inside. But that's as far as I got.... I see a pattern of (1+(1/x))^n which is the definition of e but I don't really know about this one...

3. hartnn

4 -sin x or 4 - sin n ?? if its 4- sin x, then its a constant...

4. anonymous

Nope that is not a typo. It is indeed a constant.

5. hartnn

cool! so you have the function of the form $$(1+ax)^{1/x}$$ right?

6. anonymous

How so?

7. anonymous

Okay well I can see it if you make a substitution.

8. anonymous

Go on.

9. hartnn

good, i'll tell you what we do in case of $$(1+ax)^{1/x}$$ -- > $$(1+ax)^{1/x} = [(1+ax)^{\frac{1}{ax}}]^a$$ and then use the limit formula, (if you can use) $$\lim \limits_{x\to \infty} (1+1/x)^x = e$$

10. anonymous

Brilliant! But the limit would then be 1 not 0...

11. anonymous

Which according to wolfram it's 0.

12. anonymous
13. anonymous

Ohh wait. I goofed. Have to take the ln of that.

14. anonymous

ln(1) is 0. Thank you!

15. hartnn

in the wolf, its shows ln^n

16. anonymous

Yeah it's a notational thing. That just the inside raised to the n.

17. anonymous

Kinda like sin^2(x) and (sin(x))^2.

18. hartnn

okk... i thought the answer would be 4- sin x ..

19. anonymous

Nah. It's 0.

20. hartnn

:)

21. anonymous

@hartnn : So I got up here.

22. anonymous

$\lim_{b \rightarrow 0}(1+(4-\sin(x)b)^{\frac{ 1 }{ b }}$

23. anonymous

Is that okay so far or am I way off?

24. hartnn

i assume there is ln outside of that limit and you just plugged in b =1/n

25. anonymous

Indeed sir.

26. hartnn

yes, go on

27. anonymous

Stuck >.< .

28. anonymous

Like I know that should be e but I'm having trouble relating it to the definition.

29. hartnn

whatever expression is with $$\Large 1+ ...$$ that same expression should be with $$\Large \dfrac{1}{...}$$ thats how I remember so we have 1+ (4-sin x)b so the fraction in the exponent should be $$\Large \dfrac{1}{(4-\sin x)b}$$

30. hartnn

|dw:1443855809053:dw|

31. anonymous

But the definition of e is (1+1/x)^x right? Here we have (1+(constant)b)^(1/b) . Are those equivalent?

32. hartnn
33. hartnn

34. anonymous

Interesting. I did not know that even after 4 years of calculus and differential equations lol. I learn new things every day!

35. Jhannybean

I want to learn how to solve this as well.. I get some steps but Im confused on others. :(

36. hartnn

We take that as a formula, but its easy to prove that using L'Hopital's rule.

37. hartnn

I will be writing out all the steps from the beginning

38. anonymous

Yes I know we can use L'hopital's rule but for this assignment they (Other students) can't use that.

39. hartnn

*drawing Let 4 - sin x = a , since its a constant. |dw:1443856327983:dw|

40. anonymous

Yep I got that.

41. hartnn

writing these steps for everyone's benefit :) |dw:1443856406335:dw|

42. anonymous

Yep makes sense so far...

43. hartnn

|dw:1443856506563:dw|

44. hartnn

that big bracket evaluates to 1 1^a = 1 ln 1 = 0 :D

45. anonymous

Cool!

46. anonymous

Is there somewhere I can find the proof of the stuff in the brackets?

47. Jhannybean

|dw:1443856722618:dw|

48. anonymous

Yeah to get into the proper form for the limit.

49. hartnn

lets prove it :) I'll use L'Hopitals, need to search the net for other proofs. |dw:1443856725213:dw| quick check, 0/0 form ln (1+ab) = ln 1 = 0 ab = 0 so we can apply L'Hopital's rule here

50. anonymous

Nice!

51. hartnn

we can do all kinds of mathematically legal manipulations to bring an expression in the standard form. i needed a form like (1+x)^(1/x) thats why I multiplied and divided by 'a' , which should be NON-ZERO (point to be noted.)

52. Jhannybean

oh I see I see

53. Astrophysics

How does this 0, the power would be undefined |dw:1443857378987:dw|

54. Astrophysics

Oooh wait nvm, n = 1/b when n-> infinity, b ->9

55. Astrophysics

b->0*

56. Jhannybean

Yeah, there you go

57. Astrophysics

Haha, I totally missed that, ok it's good now. Great explanation @hartnn thanks

58. Jhannybean

me 4.

59. anonymous

@hartnn

60. anonymous

|dw:1443858029848:dw|

61. hartnn

http://www.wolframalpha.com/input/?i=lim+n-%3E+infty+%281%2Ba%2Fn%29%5E%28%28n%29%29+ ln e^a = a ln e = a a = 4-sin x thats what I first got. but this wolf answer got me all confused and I ended up using b->infty instead of b->0

62. anonymous

But that's wrong though. b goes to 0, not n.

63. anonymous

not infinity*

64. Jhannybean

can you explain as to why b $$\rightarrow$$ 0 and not b $$\rightarrow \infty$$ ?

65. hartnn

true, and it makes sense logically too, constant/n = very very small no. 1+ very very small no. = 1 1^ very very larger number = 1 ln 1 =0 so the limit must go to 0 but with all the mathematical steps, I still get the answer as 4-sin x

66. anonymous

It's 0 according to wolf :( .

67. anonymous

Asking around. Like my feel is that the inside of that logarithm should be a 1.

68. anonymous

Only then can we get a 0.

69. hartnn
70. anonymous

It's witchcraft I tell you!

71. anonymous

This is the statement of t he original problem.

72. hartnn

we can only bring limit inside a function if that function is continuous. and logarithm is indeed continuous...

73. hartnn

http://www.wolframalpha.com/input/?i=lim+n-%3E+infinity+n*+ln+%281%2B%284-sin%28x%29%29%2Fn%29 even that gives 4-sin x!

74. Jhannybean

haha oh my goodness x_x

75. anonymous

Ohh wow. Wolfram is apparantly wrong.

76. anonymous
77. Jhannybean

LOL

78. Jhannybean

"which is apparently related to e" xD

79. Jhannybean
80. anonymous

Okay wow wolfram can't read notation clearly -.- .

81. hartnn

its very rare case where wolfram goes wrong

82. anonymous

Wow ._. ...

83. hartnn

$$\color{blue}{\text{Originally Posted by}}$$ @hartnn okk... i thought the answer would be 4- sin x .. $$\color{blue}{\text{End of Quote}}$$ and then wasted an hour :P

84. anonymous

All because we hail our god wolfram alpha too much ._. ...

85. Jhannybean

xD It hardly fails!!! As humans we value consistency and dependability :P