Question

Determine the limit of the sequence, and prove it directly from the
definition." ∀€>0 ∃M ∀n > M ( |an - L| < € )
an = (7n - 2)/(n^3 + 2n + 1)

I know the limit is 0, essentially I'm having trouble just simplifying the absolute value of |(7n - 2)/(n^3 + 2n + 1) - 0 | < €

Answer 1

\[\left|\frac{7n - 2}{n^3 + 2n + 1}\right|\le\left|\frac{7n}{n^3}\right|=\left|\frac{7n}{n^3}\right|=\frac{7}{n^2}\]

Answer 2

obviously skip the redundant step I have in there