1. ## 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)=$$\displaystyle \infty\cdot1=\infty But the answer in my textbook is \displaystyle 1. Am I missing something here? Help appreciated. 2. \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2} = \infty Are you sure that it's \displaystyle \lim_{n\rightarrow\infty} instead of \displaystyle \lim_{n\rightarrow 1} or something? Because the limit approaching to infinity is infinity (as you found), not 1. 3. Originally Posted by disclaimer Hello; \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2} To me, it's \displaystyle \infty, as I think we can write: \displaystyle \lim_{n\rightarrow\infty}\sqrt{n^{10}-2n^2+2}=$$\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? 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

$\displaystyle \lim_{n\rightarrow\infty} (\sqrt{n^{10}-2n^2+2} - \sqrt{n^{10} + \text{polynomial stuff of degree less than 10}}\, )$.

4. 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.

5. Could it be $\displaystyle u_n = \sqrt[n]{{n^{10} - 2n^2 + 2}}$?