anonymous
  • anonymous
limit of ( n/(n-4) )^n as n goes to infinity??
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
positive infinitiy
anonymous
  • anonymous
or wait, nvm
anonymous
  • anonymous
1

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anonymous
  • anonymous
yea..the calculator says e^4...have no idea how to get there
anonymous
  • anonymous
\[\lim_{n \rightarrow \infty} [\frac{n}{n-4}]^n\] ?
anonymous
  • anonymous
yes thats the prob
anonymous
  • anonymous
are you sure that your answer is e^4?
anonymous
  • anonymous
because I got 1 too lol
anonymous
  • anonymous
ok yea thats wat i was thinking too..thank u!
anonymous
  • anonymous
\[\lim_{n \rightarrow \infty} e^{n \ln \frac{n}{n-4}}\] 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 :)
anonymous
  • anonymous
I get \[e^4\]
anonymous
  • anonymous
how? .-.
anonymous
  • anonymous
hmm, I prolly missed out a point that I can't seem to find ._.
nowhereman
  • nowhereman
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
anonymous
  • anonymous
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.
nowhereman
  • nowhereman
\[\lim_{n→∞}{\left(\frac{n}{n-4}\right)^{n}} = \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{m+4}} = \lim_{m→∞}{\left(1+\frac{4}{m}\right)^{m}} \cdot \lim_{m→∞}{\left(\frac{m+4}{m}\right)^{4}}\]\[ = e^4 \cdot 1\]
anonymous
  • anonymous
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
anonymous
  • anonymous
You should get what nowhereman got.
anonymous
  • anonymous
/I got.
nowhereman
  • nowhereman
misha: true, but you can only pull the limit inside, if the outer part does not depend on the variable (n here)
anonymous
  • anonymous
then I was wrong, misha follow both of their ways ^_^
nowhereman
  • nowhereman
You should probably take a look at http://en.wikipedia.org/wiki/Exponential_function or something similar.
anonymous
  • anonymous
ok thanks for clarifying!
anonymous
  • anonymous
I gave you a medal sstarica
anonymous
  • anonymous
lol, thank you :) but I didn't work for it
anonymous
  • anonymous
for being a good sport :)
anonymous
  • anonymous
^_^ thank you
anonymous
  • anonymous
You can have one too, nowhereman...
anonymous
  • anonymous
lol I gave both of you a medal
anonymous
  • anonymous
loki
anonymous
  • anonymous
I prolly am just seeing it, but just to make sure, is there something wrong b/w you and nowhereman?
anonymous
  • anonymous
Not that I am aware of. Why?
anonymous
  • anonymous
I've never talked to him.
anonymous
  • anonymous
mm, lol nvm, you guys just act weird around each other?
anonymous
  • anonymous
?
anonymous
  • anonymous
lol, I'm prolly just seeing it, it's nothing ^_^
anonymous
  • anonymous
he doesn't 'chat'
anonymous
  • anonymous
noticed :)

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