find the limit of the following sequence or determine that the sequence diverges.\n{\\frac{2n^{2}-1}{7n^{2}+1…

find the limit of the following sequence or determine that the sequence diverges.\n{\\frac{2n^{2}-1}{7n^{2}+1}}\nselect the correct choice below and fill in any answer boxes to complete the choice.\na. the limit of the sequence is. (type an exact answer.)\nb. the sequence diverges.

find the limit of the following sequence or determine that the sequence diverges.\n{\\frac{2n^{2}-1}{7n^{2}+1}}\nselect the correct choice below and fill in any answer boxes to complete the choice.\na. the limit of the sequence is. (type an exact answer.)\nb. the sequence diverges.

Answer

Explanation:

Step1: Divide numerator and denominator by $n^{2}$

$\lim_{n\rightarrow\infty}\frac{2n^{2}-1}{7n^{2}+1}=\lim_{n\rightarrow\infty}\frac{2-\frac{1}{n^{2}}}{7 + \frac{1}{n^{2}}}$

Step2: Apply limit rules

As $n\rightarrow\infty$, $\lim_{n\rightarrow\infty}\frac{1}{n^{2}} = 0$. So $\lim_{n\rightarrow\infty}\frac{2-\frac{1}{n^{2}}}{7+\frac{1}{n^{2}}}=\frac{2 - 0}{7+0}$

Step3: Simplify the expression

$\frac{2 - 0}{7+0}=\frac{2}{7}$

Answer:

A. The limit of the sequence is $\frac{2}{7}$