Use the first six terms to predict the limit of the sequence $a _ { n } = \frac { 8 n ^ { 3 } + 6 } { n + 3 }$ (assume $n$ begins with 1).
A) $\frac { 8 } { 3 }$
B) 183
C) 126
D) 0
E) the sequence diverges

Question

Quizplus · Accepted Answer

To predict the limit of the sequence as $n$ approaches infinity, we examine the behavior of the sequence's formula. The dominant term in the numerator is $8n^3$ and in the denominator is $n$. As $n$ becomes very large, the $+6$ and $+3$ become negligible, so the sequence behaves like $\frac{8n^3}{n} = 8n^2$, which grows without bound. Therefore, the sequence diverges.

Use the First Six Terms to Predict the Limit of the Sequence