Question

Another limit question...

Answer 1

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

Answer 2

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...

Answer 3

4 -sin x or 4 - sin n ??

if its 4- sin x, then its a constant...

Answer 4

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

Answer 5

cool!

so you have the function of the form

\((1+ax)^{1/x}\)
right?

Answer 6

How so?

Answer 7

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

Answer 8

Go on.

Answer 9

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)

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

Answer 10

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

Answer 11

Which according to wolfram it's 0.

Answer 12

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

Answer 13

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

Answer 14

ln(1) is 0. Thank you!

Answer 15

in the wolf, its shows ln^n

Answer 16

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

Answer 17

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

Answer 18

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

Answer 19

Nah. It's 0.

Answer 20

:)

Answer 21

@hartnn : So I got up here.

Answer 22

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

Answer 23

Is that okay so far or am I way off?

Answer 24

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

Answer 25

Indeed sir.

Answer 26

yes, go on

Answer 27

Stuck >.< .

Answer 28

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

Answer 29

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