Let

$a_n=\frac{5n^2+14n}{3n^4-5n^2-22},b_n=\frac{5}{3n^2}$

Calculate the $limit.$

$(Givean$ exact answer. Use symbolic notation and fractions where needed. Enter DNE $if$ the $limit$ does not exist.$)$

$\underset{n}{lim}\frac{l}{l}$

Determine the convergence $or$ divergence $of{\displaystyle \underset{n=1}{\overset{}{}}}{a}_n.$

${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ diverges $by$ the Limit Comparison Test because $\underset{n}{lim}\frac{a_n}{b_n}is$ finite and ${\displaystyle \underset{n=1}{\overset{}{}}}{b}_n$ diverges.

$Itis$ not possible $to$ use the Limit Comparison Test $to$ determine the convergence $or$ divergence $of{\displaystyle \underset{n=1}{\overset{}{}}}{a}_n.$

${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ converges $by$ the Limit Comparison Test because $\underset{n}{lim}\frac{a_n}{b_n}is$ finite and ${\displaystyle \underset{n=1}{\overset{}{}}}{b}_n$ converges.

${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ converges $by$ the Limit Comparison Test because $\underset{n}{lim}\frac{a_n}{b_n}is$ finite and ${\displaystyle \underset{n=1}{\overset{}{}}}{b}_n$ diverges.