Epsilon proof of a sequence's limit - algebra issues

by dugy1001   Last Updated August 12, 2018 02:20 AM - source

I am attempting to prove $\lim_{k\to\infty}\dfrac{k^2}{k^2+2k+2}=1$. However, I am getting tripped up on the algebra. I believe I want to show that there exists some $N\in\mathbb{N}$ such that when $k\geq N$, $\left|\dfrac{k^2}{k^2+2k+2}-1\right|<\epsilon$ for arbitrary $\epsilon>0$. I set out to find a choice for $N$ by turning $1$ into $\dfrac{k^2+2k+2}{k^2+2k+2}$, but I am not sure where to go from $\left|\dfrac{-2k-2}{k^2+2k+2}\right|<\epsilon$. How can I rearrange this inequality to be in terms of $k$? Thank you!

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Updated June 20, 2018 16:20 PM