Can you Find the limit of the sequence or determine that the limit does not exist for the sequence {n^4/(n^5+1)}?

1 Answer
Mar 12, 2018

The sequence has the same behaviour as ${n}^{4} / {n}^{5} = \frac{1}{n}$ when $n$ is large

Explanation:

You should manipulate the expression just a bit to make that statement above clear. Divide all terms by ${n}^{5}$.

${n}^{4} / \left({n}^{5} + 1\right) = \frac{{n}^{4} / {n}^{5}}{\frac{{n}^{5} + 1}{n} ^ 5} = \frac{\frac{1}{n}}{1 + \frac{1}{n} ^ 5}$. All these limits exist when $n \to \infty$, so we have:

${\lim}_{n \to \infty} {n}^{4} / \left({n}^{5} + 1\right) = \frac{{n}^{4} / {n}^{5}}{\frac{{n}^{5} + 1}{n} ^ 5} = \frac{\frac{1}{n}}{1 + \frac{1}{n} ^ 5} = \frac{0}{1 + 0} = 0$, so the sequence tends to 0