Thread: Limit for a quite simple function

1. Limit for a quite simple function

Hi!
I have a homework and I could not figure out 1 of 10 tasks.
I think it is really simple, if I take the "brutal" method I can see that function goes to -inf.
Even with a grapher I can see that function goes to -inf. (except for if x < 0).
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Only the first equation.
As far as I figured out I can do many things here, but the result... not so right.
First I tried to simplify, (n^4 + 1)(n^2-1).
Then tried to multiply and divide the equation with sqrt(...) + sqrt(...), tried to get a^2 - b^2 = (a-b)(a+b)
When I simplified the upper part I get n^2 - n^4, since there is a n to divide, then I can divide everything with n and get n - n^3.
Then inserting inf. I get (inf. - inf.), so no good. Tried to divide everything with the highest n, which is n^3.
Since n^3/n^3 is 1 and 1/n^3 is inf., I get -1/2= -0.5
I don't know where is the mistake!
Can someone help?

2. Re: Limit for a quite simple function

What you're doing is good, if you write the steps:
$\lim_{n\to \infty} \frac{\sqrt{(n^4+1)(n^2-1)}-\sqrt{n^6-1}}{n}=\lim_{n\to \infty}\frac{n^2-n^4}{n(\sqrt{(n^4+1)(n^2-1)}-\sqrt{n^6-1})}$ $=\lim_{n\to \infty} \frac{n-n^3}{\sqrt{(n^4+1)(n^2-1)}-\sqrt{n^6-1}}$ $=\lim_{n\to \infty} \frac{n^3\left(\frac{1}{n^2}-1\right)}{\sqrt{1+\frac{1}{n^4}-\frac{1}{n^2}-\frac{1}{n^6}}+\sqrt{1-\frac{1}{n^6}}}=\frac{-1}{2}$

3. Re: Limit for a quite simple function

Yeah, I got the same result the same way, but have you seen the graph? It is like a parabola upside down and with a part missing, but it goes mostly to -inf.
So I don't know if the result is correct.

4. Re: Limit for a quite simple function

A confirmation of wolphram alpha:
lim_&#40;n to infty&#41; &#40;sqrt&#40;&#40;n&#94;4&#43;1&#41;&#40;n&#94 ;2-1&#41;&#41;-sqrt&#40;n&#94;6-1&#41;&#41;&#47;&#40;n&#41; - Wolfram|Alpha

Yes, I saw the graph and you'll see the graph approaches to $\frac{-1}{2}$ if $n\to +\infty$ and to $\frac{1}{2}$ if $n\to -\infty$.
graph &#40;sqrt&#40;&#40;x&#94;4&#43;1&#41;&#40;x&#94 ;2-1&#41;&#41;-sqrt&#40;x&#94;6-1&#41;&#41;&#47;&#40;x&#41; - Wolfram|Alpha

5. Re: Limit for a quite simple function

Oh, I needed to double check the equation!
Did input something wrong and gave that weird graph.
Thanks for clearing that up!

6. Re: Limit for a quite simple function

You're welcome!