# Thread: help finding a limit

1. ## help finding a limit

Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
Thank you

2. Originally Posted by cowboys111
Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
Thank you
I'm not sure how to prove it aside from just plugging in really large numbers, but $\displaystyle \lim_{k \to \infty} k^{1/k} = 1$.

3. Originally Posted by cowboys111
Hi, can someone show me how to find this limit i cant figure it out. Lim as k goes to infinity (k^(1/k))/5 ?
Thank you
$\displaystyle L=\lim_{k\to\infty}\frac{\sqrt[k]{k}}{5}$

Taking the natural log of both sides gives

$\displaystyle \ln(L)=\lim_{k\to\infty}\frac{\ln(k)}{k}-\ln(5)$

Since this yields an indeterminate form we apply L'hopitals rule go get

$\displaystyle \ln(L)=\lim_{k\to\infty}\frac{\frac{1}{k}}{1}-\ln(5)=-\ln(5)$

So $\displaystyle L=e^{-\ln(5)}=\frac{1}{5}$

4. Originally Posted by icemanfan
I'm not sure how to prove it aside from just plugging in really large numbers, but $\displaystyle \lim_{k \to \infty} k^{1/k} = 1$.
Try considering

$\displaystyle \forall{x}>1,1\leq{x^{\frac{1}{x}}}\leq\bigg(1+\fr ac{1}{\sqrt{x}}\bigg)$

5. thank you