Question

limit of ( n/(n-4) )^n as n goes to infinity??

Answer 1

positive infinitiy

Answer 2

or wait, nvm

Answer 3

1

Answer 4

yea..the calculator says e^4...have no idea how to get there

Answer 5

\[\lim_{n \rightarrow \infty} [\frac{n}{n-4}]^n\]

?

Answer 6

yes thats the prob

Answer 7

are you sure that your answer is e^4?

Answer 8

because I got 1 too lol

Answer 9

ok yea thats wat i was thinking too..thank u!

Answer 10

\[\lim_{n \rightarrow \infty} e^{n \ln \frac{n}{n-4}}\]

first you find the limit of the fraction :

\[\lim_{n \rightarrow \infty} \frac{n}{n-4} = \lim_{n \rightarrow \infty} \frac{n}{n} = 1\]

so now plug this value in the first equation I wrote:

\[\lim_{n \rightarrow \infty} e^0 = 1\]

np :)

Answer 11

I get \[e^4\]

Answer 12

how? .-.

Answer 13

hmm, I prolly missed out a point that I can't seem to find ._.

Answer 14

No, that is not how it works. This way there wouldn't be any constant e. The problem with your approach is, that when the ln goes to zero, at the same time n goes to infinity and so you can't say that their product goes to zero!
But you can do the following substitution:
\[\lim_{n→∞}{\left(\frac{n}{n-4}\right)^{n}} = \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{m+4}} = \lim_{m→∞}{\left(1+\frac{4}{m}\right)^{m}} \cdot \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{4}} = e^4 \cdot 1

Answer 15

Set the expression equal to y and take the log of both sides. You'll end up with\[\log y = n \log \frac{n}{n-4}=n \log \frac{1}{1-4/n}=-n \log (1-4/n)=-\frac{\log (1-4/n)}{1/n}\]That's now indeterminate, so you can apply L'Hopital's and take the limit. Then undo the log.

Answer 16

\[\lim_{n→∞}{\left(\frac{n}{n-4}\right)^{n}} = \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{m+4}} = \lim_{m→∞}{\left(1+\frac{4}{m}\right)^{m}} \cdot \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{4}}\]\[ = e^4 \cdot 1\]

Answer 17

when the degree is same on the top and bottom, dont you jus look at the coeffecients of the highest degree which is 1/1= 1

Answer 18

You should get what nowhereman got.

Answer 19

/I got.

Answer 20

misha: true, but you can only pull the limit inside, if the outer part does not depend on the variable (n here)

Answer 21

then I was wrong, misha follow both of their ways ^_^

Answer 22

You should probably take a look at http://en.wikipedia.org/wiki/Exponential_function or something similar.

Answer 23

ok thanks for clarifying!

Answer 24

I gave you a medal sstarica

Answer 25

lol, thank you :) but I didn't work for it

Answer 26

for being a good sport :)

Answer 27

^_^ thank you

Answer 28

You can have one too, nowhereman...

Answer 29

lol I gave both of you a medal

Answer 30

loki

Answer 31

I prolly am just seeing it, but just to make sure, is there something wrong b/w you and nowhereman?

Answer 32

Not that I am aware of. Why?

Answer 33

I've never talked to him.

Answer 34

mm, lol nvm, you guys just act weird around each other?

Answer 35

?

Answer 36

lol, I'm prolly just seeing it, it's nothing ^_^

Answer 37

he doesn't 'chat'

Answer 38

noticed :)