Use the first six terms to predict the limit of the sequence $ a_{n}=\frac{5 n^{3}}{n^{3}+3} $ (assume $ n $ begins with 1).
A) 4
B) 5
C) $ \frac{23}{6} $
D) $ \frac{32}{5} $
E) the sequence diverges

Question

Quizplus · Accepted Answer

The sequence $ a_{n}=\frac{5 n^{3}}{n^{3}+3} $ approaches 5 as $ n $ becomes very large, because the $ n^{3} $ term in the numerator and denominator dominates, making the fraction approach $ \frac{5n^{3}}{n^{3}} = 5 $.

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