Question

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

Answer 1

The sequence has the same behaviour as `n^4/n^5 = 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/(n^5+1) = (n^4/n^5)/((n^5+1)/n^5)= (1/n)/(1+1/n^5)`. All these limits exist when `n->oo`, so we have:

`lim_(n->oo)n^4/(n^5+1) = (n^4/n^5)/((n^5+1)/n^5)= (1/n)/(1+1/n^5) = 0/(1+0)=0`, so the sequence tends to 0