# strange... limit

• Mar 8th 2008, 01:01 AM
disclaimer
strange... limit
Hello;

$\displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2}$

To me, it's $\displaystyle \infty$, as I think we can write:

$\displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2}=$$\displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}\left(1-\frac{2}{n^8}+\frac{2}{n^{10}}\right)}=$$\displaystyle \lim_{n\rightarrow\infty}\left(n^5\sqrt{1-\frac{2}{n^8}+\frac{2}{n^{10}}}\right)=$$\displaystyle \infty\cdot1=\infty But the answer in my textbook is \displaystyle 1. Am I missing something here? :confused: Help appreciated. :) • Mar 8th 2008, 01:41 AM wingless \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2} = \infty Are you sure that it's \displaystyle \lim_{n\rightarrow\infty} instead of \displaystyle \lim_{n\rightarrow 1} or something? Because the limit approaching to infinity is infinity (as you found), not 1. • Mar 8th 2008, 01:41 AM mr fantastic Quote: Originally Posted by disclaimer Hello; \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2} To me, it's \displaystyle \infty, as I think we can write: \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2}=$$\displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}\left(1-\frac{2}{n^8}+\frac{2}{n^{10}}\right)}=$$\displaystyle \lim_{n\rightarrow\infty}\left(n^5\sqrt{1-\frac{2}{n^8}+\frac{2}{n^{10}}}\right)=$$\displaystyle \infty\cdot1=\infty$

But the answer in my textbook is $\displaystyle 1$. Am I missing something here? :confused: Mr F says: No. But your book probably is ..... the print for the continuation of the expression that you're taking the limit of.

Help appreciated. :)

The question was probably meant to be something like

$\displaystyle \lim_{n\rightarrow\infty} (\sqrt{n^{10}-2n^2+2} - \sqrt{n^{10} + \text{polynomial stuff of degree less than 10}}\, )$.
• Mar 8th 2008, 01:50 AM
disclaimer
Well, in the book they give just $\displaystyle u_n=\sqrt{n^{10}-2n^2+2}$ and the problem is to find the limit of the sequence having such a general formula.

Thanks for taking the time to look into that. :)
• Mar 8th 2008, 04:01 AM
Plato
Could it be $\displaystyle u_n = \sqrt[n]{{n^{10} - 2n^2 + 2}}$?