Find the limit of the following sequence or determine that the sequence diverges.

Select the correct choice below and fill in any answer boxes to complete the choice.
A. The limit of the sequence is . (Type an exact answer.)
B. The sequence diverges.

Ask by Turnbull Guerrero. in the United States

Mar 15,2025

Question

Find the limit of the following sequence or determine that the sequence diverges.

Select the correct choice below and fill in any answer boxes to complete the choice.
A. The limit of the sequence is . (Type an exact answer.)
B. The sequence diverges.

Ask by Turnbull Guerrero. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

Evaluate the limit by following steps:
- step0: Evaluate using transformations:
  $$\lim _{nightarrow +\infty}\left(\frac{n^{3}}{n^{4}+1}ight)$$
- step1: Rewrite the expression:
  $$\lim _{nightarrow +\infty}\left(\frac{n^{3}}{n^{4}}	imes \frac{1}{1+\frac{1}{n^{4}}}ight)$$
- step2: Calculate:
  $$\lim _{nightarrow +\infty}\left(\frac{1}{n}	imes \frac{1}{1+\frac{1}{n^{4}}}ight)$$
- step3: Rewrite the expression:
  $$\lim _{nightarrow +\infty}\left(\frac{1}{n}ight)	imes \lim _{nightarrow +\infty}\left(\frac{1}{1+\frac{1}{n^{4}}}ight)$$
- step4: Calculate:
  $$\lim _{nightarrow +\infty}\left(\frac{1}{n}ight)	imes \frac{\lim _{nightarrow +\infty}\left(1ight)}{\lim _{nightarrow +\infty}\left(1+\frac{1}{n^{4}}ight)}$$
- step5: Rewrite the expression:
  $$\lim _{nightarrow +\infty}\left(\frac{1}{n}ight)	imes \frac{1}{\lim _{nightarrow +\infty}\left(1+\frac{1}{n^{4}}ight)}$$
- step6: Rewrite the expression:
  $$\lim _{nightarrow +\infty}\left(\frac{1}{n}ight)	imes \frac{1}{1}$$
- step7: Rewrite the expression:
  $$0	imes \frac{1}{1}$$
- step8: Calculate:
  $$0	imes 1$$
- step9: Calculate:
  $$0$$
The limit of the sequence $$ \left\{\frac{n^{3}}{n^{4}+1}ight\} $$ as $$ n $$ approaches infinity is $$ 0 $$.

So, the correct choice is:
A. The limit of the sequence is $$ 0 $$.
The limit of the sequence is 0.

Find the limit of the following sequence or determine that the sequence diverges.

Select the correct choice below and fill in any answer boxes to complete the choice.
A. The limit of the sequence is . (Type an exact answer.)
B. The sequence diverges.

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Find the limit of the following sequence or determine that the sequence diverges. Select the correct choice below and fill in any answer boxes to complete the choice. A. The limit of the sequence is . (Type an exact answer.) B. The sequence diverges.

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Find the limit of the following sequence or determine that the sequence diverges.

Select the correct choice below and fill in any answer boxes to complete the choice.
A. The limit of the sequence is . (Type an exact answer.)
B. The sequence diverges.