Question

limit of an = (2n^2)/(n^2+6) ?

Answer 1

\[
\n^2+6\approx n^2
\]
near infinity
So how much is the limit?

Answer 2

\[
n^2+6 \approx n^2
\]

Answer 3

So
\[
\frac {2 n^2}{n^2+6}\approx \frac {2 n^2}{n^2}=2
\]
the limit is 2

Answer 4

Another way is to do the following
\[

\frac{2 n^2}{n^2\left( 1 +\frac 6 {n^2} \right ) }=\frac{2 }{\left( 1 +\frac 6 {n^2} \right ) }
\]
It is easy now to see that the limit is 2