Use definition of limit to prove.. Question: Let $x_n = 1+\frac{5+n}{n^2}$. Use the definition to prove that 
$$\lim_{x \to \infty} x_n = 1$$
I know from the definition, that $|x_n-x|<\epsilon$, so I start with:
$$|1+ \frac{5+n}{n^2} -1| = |\frac{5+n}{n^2}| <\epsilon$$
Then I do a bunch of algebra and I can't figure out a closed form solution for $n$. help!

Hint: For $n\ge 5,$ you have $\frac{5+n}{n^2}\le\frac{2n}{n^2}=\frac2n.$ How large must $n$ be in order that $\frac2n<\epsilon$? How can you ensure that $n$ is this large and also $\ge 5$?
