Question

Find the limit of the sequence or determine that the limit does not exist

(2n^3 -1 ) / (8n^3 +1). Also mention if the sequence is monotonic or not and if it is bounded or unbounded. thankyou

Find the limit of the sequence or determine that the limit does not exist 203-1 8n3 +1

Answer 1

Yes, sequence is monotonic, increasing

Yes, it is bounded to 1/4

$\dpi{300} \fn_phv \tiny \begin{align*} & \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }\frac{2n^{3}-1}{8n^{3}+1} \\ & =\lim _{n\to \infty }\frac{2-\frac{1}{n^{3}}}{8+\frac{1}{n^{3}}} \\ & =\frac{2-0}{8+0} \\ & =\frac{2}{8} \\ & =\frac{1}{4} \end{align}$

limit of the sequence is 1/4