# Thread: limits of sequences with ln(n)

1. ## limits of sequences with ln(n)

I have a problem with these sequences: $\lim_{x\to\infty} \frac{ln(3n^2+20n+5)}{ln(n^9-3n+12)}$
$\lim_{x\to\infty} n ln(n^{2}+1)-2n(lnn)(lnn)^{1/n}$
$\lim_{x\to\infty}\frac{1*4*7*...*(3n+1)}{2*5*8*... *(3n+2)}$

The first one - I don't know how to get rid of the ln(something) function or maybe I'm just doing it in a wrong way. The second one - everything would be great if there wasn't this $ln(n)[ln(n)]^{1/n}$ which I don't know what to do with. The third one - I don't have any idea about this one... I would appreciate your hints or solutions. I've been trying to solve them for so long but I constantly get wrong answers (I used wolfram alpha).

2. ## Re: limits of sequences with ln(n)

Originally Posted by Lisa91
I have a problem with these sequences: lim n-> infinity [ln(3n^2+20n+5)]/[ln(n^9-3n+12)] lim n-> ininity (n ln(n^2+1)-2n(lnn)(lnn)^(1/n)) lim n> infinity (1*4*7*...*(3n+1))/(2*5*8*...*(3n+2)) The first one - I don't know how to get rid of the ln(something) function or maybe I'm just doing it in a wrong way. The second one - everything would be great if there wasn't this lnn)(lnn)^(1/n) which I don't know what to do with. The third one - I don't have any idea about this one... I would appreciate your hints or solutions. I've been trying to solve them for so long but I constantly get wrong answers (I used wolfram alpha).
I wonder why you do not learn to post is a readable format?

LaTex Code is easy to learn.

3. ## Re: limits of sequences with ln(n)

Sorry, I've just done it

4. ## Re: limits of sequences with ln(n)

That is a lot more readable.

For the first one, I think you can use L'Hopital's rule.

- Hollywood

5. ## Re: limits of sequences with ln(n)

Are you sure you're making x approach infinity?

6. ## Re: limits of sequences with ln(n)

Wolframalpha says that the second one is $-\infty$ and the third one is zero. On the third one, I think it might help to take the log so you have the sum of $\log{\frac{3k+1}{3k+2}}$.

- Hollywood

7. ## Re: limits of sequences with ln(n)

Oh, yes, obviously, it's n which approaches infinity. I'm done with the third one but I can't do the first and the second one.

8. ## Re: limits of sequences with ln(n)

Hello

1. Using l'Hôpital:

$\\\lim_{n\to\infty}\dfrac{\ln(3n^2+20n+5)}{\ln(n^9-3n+12)}=\dfrac{\lim_{n\to\infty}\dfrac{6n+20}{3n^2 +20n+5}}{\lim_{n\to\infty}\dfrac{9n^8+3}{n^9-3n+12}}\\\lim_{n\to\infty}\dfrac{\ln(3n^2+20n+5)}{ \ln(n^9-3n+12)}=\lim_{n\to\infty}\dfrac{(6n+20)(n^9-3n+12)}{(9n^8+3)(3n^2+20n+5)}$

You can apply l'Hôpital few more times, but you know that:

$\\\lim_{n\to\infty}\dfrac{\ln(3n^2+20n+5)}{\ln(n^9-3n+12)}=\lim_{n\to\infty}\dfrac{6n^{10}+\ldots}{27 n^{10}+\ldots}\\\lim_{n\to\infty}\dfrac{\ln(3n^2+2 0n+5)}{\ln(n^9-3n+12)}=\dfrac{6}{27}=\dfrac{2}{9}$

2. Making this change of variable:

$\\n=\dfrac{1}{t}\\\lim_{x\to\infty}n\ln(n^2+1)-2n(\ln(n))(\ln(n))^{\frac{1}{n}}=\lim_{t\to0} \dfrac{1}{t}\ln{}\left({}\frac{1}{t^2}{}+1\right)-2\dfrac{1}{t}\ln\left(\frac{1}{t}\right){}\left({} \ln{}\left({}\frac{1}{t}\right)\right)^t\\=\lim_{t {}\to{}0}\dfrac{\ln\left(\frac{1}{n^2}+1\right)-2\ln\left(\frac{1}{n}\right)^{t+1}}{t}$

You can continue, just applies L'hopital (because is $\dfrac{0}{0}$, before that it was $\infty-\infty$) many times.... The answer is $-\infty$

I can't write you the full solution because I don't know how to use $\mathrm{\LaTeX}$ well....

Greetings

9. ## Re: limits of sequences with ln(n)

Originally Posted by Plato
I wonder why you do not learn to post is a readable format?

LaTex Code is easy to learn.
Are you serious?

Originally Posted by Prove It
Are you sure you're making x approach infinity?
lol I'm sure he wanted to type $n$.

10. ## Re: limits of sequences with ln(n)

Originally Posted by darthjavier
Are you serious?
Yes! I was quite serious.
If you look at the OP before it was edited, you would why.

And LaTeX is easy to learn.
If I had my way, all regular posters should be required to us LaTeX code.

11. ## Re: limits of sequences with ln(n)

Originally Posted by Plato
Yes! I was quite serious.
If you look at the OP before it was edited, you would why.
Yes, I'm agree with you. I discovered recently $\mathrm{\LaTeX}$, and I try to practice every day because is very useful.
But I mean that $\mathrm{\LaTeX}$ is difficult to learn, especially on this page. When I post it's too annoying see the equations combined with the HTML code. I can't see the HTML when I'm editing my $\mathrm{\LaTeX}$ code, I don't know why. In other pages I use an editor (in my case use TextMaker) and I only had to copy-paste. But here I have to write everything on one line, and it seems that LaTeX doesn't work well here, sometimes there are mistakes, I have to write "{}" all the time. I don't know if I'm the only one with this problems.
Thank you very much, guys. I am wondering whether I am allowed to apply the de L'Hospital rule because we have a sequence, not a function here. I think I need to proof that let's say $ln(n^2+20n+5)$ and $ln(n^9-3n+12)$ are differentiable. Is there anyone who's studying math?