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

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\lim_{n \rightarrow \infty}\ln\left( 1+ \frac{ 4\sin(x) }{ n } \right)^n\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0My 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...

hartnn
 one year ago
Best ResponseYou've already chosen the best response.44 sin x or 4  sin n ?? if its 4 sin x, then its a constant...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Nope that is not a typo. It is indeed a constant.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4cool! so you have the function of the form \((1+ax)^{1/x}\) right?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Okay well I can see it if you make a substitution.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4good, 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) \(\lim \limits_{x\to \infty} (1+1/x)^x = e\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Brilliant! But the limit would then be 1 not 0...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Which according to wolfram it's 0.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0http://www.wolframalpha.com/input/?i=lim+n%3E+infinity+ln%281%2B%284sin%28x%29%29%2Fn%29%5En

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ohh wait. I goofed. Have to take the ln of that.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ln(1) is 0. Thank you!

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4in the wolf, its shows ln^n

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yeah it's a notational thing. That just the inside raised to the n.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Kinda like sin^2(x) and (sin(x))^2.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4okk... i thought the answer would be 4 sin x ..

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@hartnn : So I got up here.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\lim_{b \rightarrow 0}(1+(4\sin(x)b)^{\frac{ 1 }{ b }}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Is that okay so far or am I way off?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4i assume there is ln outside of that limit and you just plugged in b =1/n

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Like I know that should be e but I'm having trouble relating it to the definition.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4whatever expression is with \(\Large 1+ ...\) that same expression should be with \(\Large \dfrac{1}{...}\) thats how I remember so we have 1+ (4sin x)b so the fraction in the exponent should be \(\Large \dfrac{1}{(4\sin x)b}\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0But the definition of e is (1+1/x)^x right? Here we have (1+(constant)b)^(1/b) . Are those equivalent?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4http://www.wolframalpha.com/input/?i=lim+x%3E+infty+%281%2Bax%29%5E%281%2F%28ax%29%29+

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4that will clear your doubt

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Interesting. I did not know that even after 4 years of calculus and differential equations lol. I learn new things every day!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I want to learn how to solve this as well.. I get some steps but Im confused on others. :(

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4We take that as a formula, but its easy to prove that using L'Hopital's rule.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4I will be writing out all the steps from the beginning

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yes I know we can use L'hopital's rule but for this assignment they (Other students) can't use that.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4*drawing Let 4  sin x = a , since its a constant. dw:1443856327983:dw

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4writing these steps for everyone's benefit :) dw:1443856406335:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yep makes sense so far...

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4that big bracket evaluates to 1 1^a = 1 ln 1 = 0 :D

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Is there somewhere I can find the proof of the stuff in the brackets?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1443856722618:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yeah to get into the proper form for the limit.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4lets 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

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4we 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 NONZERO (point to be noted.)

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0How does this 0, the power would be undefined dw:1443857378987:dw

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0Oooh wait nvm, n = 1/b when n> infinity, b >9

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.0Haha, I totally missed that, ok it's good now. Great explanation @hartnn thanks

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0dw:1443858029848:dw

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4http://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 = 4sin 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0But that's wrong though. b goes to 0, not n.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can you explain as to `why` b \(\rightarrow\) 0 and not b \(\rightarrow \infty\) ?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4true, 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 4sin x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It's 0 according to wolf :( .

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Asking around. Like my feel is that the inside of that logarithm should be a 1.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Only then can we get a 0.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4inside of a logarithm http://www.wolframalpha.com/input/?i=lim+n%3E+infty+%281%2B%284sin+x%29%2Fn%29%5E%28%28n%29%29+

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0It's witchcraft I tell you!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This is the statement of t he original problem.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4we can only bring limit inside a function if that function is continuous. and logarithm is indeed continuous...

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4http://www.wolframalpha.com/input/?i=lim+n%3E+infinity+n*+ln+%281%2B%284sin%28x%29%29%2Fn%29 even that gives 4sin x!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0haha oh my goodness x_x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ohh wow. Wolfram is apparantly wrong.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0"which is apparently related to e" xD

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Okay wow wolfram can't read notation clearly . .

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4its very rare case where wolfram goes wrong

hartnn
 one year ago
Best ResponseYou've already chosen the best response.4\(\color{blue}{\text{Originally Posted by}}\) @hartnn okk... i thought the answer would be 4 sin x .. \(\color{blue}{\text{End of Quote}}\) and then wasted an hour :P

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0All because we hail our god wolfram alpha too much ._. ...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0xD It hardly fails!!! As humans we value consistency and dependability :P
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