How to show convergence of a sequence?

Have you come across algebra of limits? This may prove to be useful

I have this question and other similar ones and I'm getting stuck.

Does

$\displaystyle a_n=\frac{2+7n(-1)^n+n^2}{1+n^2}$ converge?

If so find the limit.

I am having real trouble with these types of questions I know about the sandwich theorem and trying to show $\displaystyle a_n$ is unbounded therefore divergent but I'm very bad.

Please help.

From the point of a "method mathematician" and not a purist

if you divide top and bottom by n²

then you get terms of O(1/n) and O(1/n²) which tend to zero as n tends to infinity

the limit is 1

$\displaystyle \frac{2-7n+n^2}{1+n^2} \leqslant a_n \leqslant \frac{2+7n+n^2}{1+n^2}$

Then dividing the fractions on both sides (top and bottom) by $\displaystyle n^2$ and applying the AOL for sums products and quotients we get.

$\displaystyle 1 \leqslant a_n \leqslant 1$ and so $\displaystyle a_n \rightarrow 1$

Is this right?

These questions are (almost) universally expected to be done from a "first principles" pure point of view, in which case that isn't going to be acceptable. (Although it would be perfectly acceptable in a pure course at postgrad level ironically).