anonymous
  • anonymous
find the limit of k^(1/k) as k --> infinity
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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IrishBoy123
  • IrishBoy123
hint # 1 : stuff some numbers (let k = 100, 100, 10,000] into your calculator and see what you get. it is not \(\infty\) hint #2: take logs and see if you can use L'Hopital. i think you can. [taking logs will get rid of that rather ugly exponent.] so run with \(ln \ y = ln(k^{1/k})\)
anonymous
  • anonymous
yes i just plugged in some #s and the lim is definitely not infinity... I'll have to use L'H but I'm stuck with ln(K^(1/k)) now... do I take the e^ ln(y) and e^ln(k^(1/k))...
anonymous
  • anonymous
I have the lim ln k^(1/k) ==> lim ln^(k) / k ...what's the derivative of ln^k?

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IrishBoy123
  • IrishBoy123
on RHS you should have \(ln(k^{1/k}) = \frac{1}{k} ln \ k\ = \frac{ln \ k}{k} \). that is l'Hopital territory. incidentally, after plugging in some numbers, what do you think the limit is? it is much easier when you know that....
anonymous
  • anonymous
0
IrishBoy123
  • IrishBoy123
not sure which question you are now answering :p if it is as \(k \rightarrow \ \infty, \ k^{(1/k)} = 0\), you need to try again. e.g.: \(100^{(0.01)}\) = ???? for \(\frac{ln \ k}{k}\), you should see that as \(k \rightarrow \infty\), you have \(\frac{ln \ \infty}{\infty} = \frac{\infty}{\infty}\), so you can use L'Hopital.
anonymous
  • anonymous
\[\lim k ^{1/k} -> \lim \ln k ^{1/k} = \frac{ 1 }{ k }\ln k ->L'H = \lim \frac{ \frac{ 1 }{ k } }{ 1 }=\frac{ 1 }{k }\]
perl
  • perl
now plug in infinity
anonymous
  • anonymous
as k --> infinity, 1/k --> 0
anonymous
  • anonymous
yes i know but if i plug in infinity to the original equation, k^(1/k) = 1
perl
  • perl
if you plug in original equation you get an indeterminate form \( \infty ^0 \)
anonymous
  • anonymous
hmm, true. that's why I had to do L'H. Thanks
IrishBoy123
  • IrishBoy123
recap: we made the LHS ln(y) and we want to know what **y** is as \(k \rightarrow \infty\) you had this licked when you said: "as k --> infinity, 1/k --> 0". that was L'Hopital applied to the RHS. which means the LHS is what?!?!? IOW, problem solved!
anonymous
  • anonymous
oh. to get rid of ln, I need to raise e^(ln(y)) which means I need to do e^(1/k) on the RHS..which is e^0 = 1..
anonymous
  • anonymous
That took me a long time to get...but finally. Thanks everyone! The problem makes senses now.
IrishBoy123
  • IrishBoy123
well done!

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