Another limit question...

- anonymous

Another limit question...

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- anonymous

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

- 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...

- hartnn

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

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## More answers

- anonymous

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

- hartnn

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

- anonymous

How so?

- anonymous

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

- anonymous

Go on.

- 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)
\(\lim \limits_{x\to \infty} (1+1/x)^x = e\)

- anonymous

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

- anonymous

Which according to wolfram it's 0.

- anonymous

http://www.wolframalpha.com/input/?i=lim+n-%3E+infinity+ln%281%2B%284-sin%28x%29%29%2Fn%29%5En

- anonymous

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

- anonymous

ln(1) is 0. Thank you!

- hartnn

in the wolf, its shows ln^n

- anonymous

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

- anonymous

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

- hartnn

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

- anonymous

Nah. It's 0.

- hartnn

:)

- anonymous

@hartnn : So I got up here.

- anonymous

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

- anonymous

Is that okay so far or am I way off?

- hartnn

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

- anonymous

Indeed sir.

- hartnn

yes, go on

- anonymous

Stuck >.< .

- anonymous

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

- 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
\(\Large \dfrac{1}{(4-\sin x)b}\)

- hartnn

|dw:1443855809053:dw|

- anonymous

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

- hartnn

http://www.wolframalpha.com/input/?i=lim+x-%3E+infty+%281%2Bax%29%5E%281%2F%28ax%29%29+

- hartnn

that will clear your doubt

- anonymous

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

- Jhannybean

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

- hartnn

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

- hartnn

I will be writing out all the steps from the beginning

- anonymous

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

- hartnn

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

- anonymous

Yep I got that.

- hartnn

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

- anonymous

Yep makes sense so far...

- hartnn

|dw:1443856506563:dw|

- hartnn

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

- anonymous

Cool!

- anonymous

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

- Jhannybean

|dw:1443856722618:dw|

- anonymous

Yeah to get into the proper form for the limit.

- 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

- anonymous

Nice!

- 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.)

- Jhannybean

oh I see I see

- Astrophysics

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

- Astrophysics

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

- Astrophysics

b->0*

- Jhannybean

Yeah, there you go

- Astrophysics

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

- Jhannybean

me 4.

- anonymous

@hartnn

- anonymous

|dw:1443858029848:dw|

- 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

- anonymous

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

- anonymous

not infinity*

- Jhannybean

can you explain as to `why` b \(\rightarrow\) 0 and not b \(\rightarrow \infty\) ?

- 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

- anonymous

It's 0 according to wolf :( .

- anonymous

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

- anonymous

Only then can we get a 0.

- hartnn

inside of a logarithm
http://www.wolframalpha.com/input/?i=lim+n-%3E+infty+%281%2B%284-sin+x%29%2Fn%29%5E%28%28n%29%29+

- anonymous

It's witchcraft I tell you!

- anonymous

This is the statement of t he original problem.

##### 1 Attachment

- hartnn

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

- 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!

- Jhannybean

haha oh my goodness x_x

- anonymous

Ohh wow. Wolfram is apparantly wrong.

- anonymous

http://math.stackexchange.com/questions/1462100/evaluating-the-limit-lim-n-rightarrow-infty-ln-left-1-frac-4-sinx

- Jhannybean

LOL

- Jhannybean

"which is apparently related to e" xD

- Jhannybean

http://i.imgur.com/F4xr6yp.png

- anonymous

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

- hartnn

its very rare case where wolfram goes wrong

- anonymous

Wow ._. ...

- 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

- anonymous

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

- Jhannybean

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

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