Rigorous Limits

**I-Think** · September 30th 2010, 02:41 PM

We've begun limits, but the more rigorous approach I have not fully understood yet, so I'm hoping someone could help me please

Question
$lim_{n\rightarrow{\infty}} \frac{n^2+1}{n^2+2}$

Answer
Determine some real number $L$ such that
$|\frac{n^2+1}{n^2+2}-L|<\epsilon$

Rough work(aka scrap work)
BY intuitive guess, L=1. So we need
$|\frac{n^2+1}{n^2+2}-1|<\epsilon$

We need a $N$ such that $\frac{n^2+1}{n^2+2}<\epsilon+1$

Can I choose $N>\frac{1}{1+\epsilon}$ , (by Archimedean property),
invert to obtain $\frac{1}{N}<\epsilon+1$
and proceed from there?

**Plato** · September 30th 2010, 02:46 PM

I suggest that you use $\left| {\dfrac{{n^2 + 1}}{{n^2 + 2}} - 1} \right| = \left| {\dfrac{{ - 1}}{{n^2 + 2}}} \right|$

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