Problem

\( \lim _{n \rightarrow \infty} \frac{n^{2}\left(\sin n+\cos ^{3} n\right)}{\left(n^{2}+1\right)(n-3)} \)

Answer

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Answer

\(\lim _{n \rightarrow \infty} \frac{n^{2}\left(\sin n+\cos ^{3} n\right)}{\left(n^{2}+1\right)(n-3)} = 0 + 0\)

Steps

Step 1 ：\( \lim _{n \rightarrow \infty} \frac{n^{2}\left(\sin n+\cos ^{3} n\right)}{\left(n^{2}+1\right)(n-3)} = \frac{\infty}{\infty} \)

Step 2 ：\( \lim _{n \rightarrow \infty} \frac{n^{2} \sin n}{(n^{2}+1)(n-3)} + \lim _{n \rightarrow \infty} \frac{n^{2} \cos^{3} n}{(n^{2}+1)(n-3)}\)

Step 3 ：\(\lim _{n \rightarrow \infty} \frac{n^{2}\left(\sin n+\cos ^{3} n\right)}{\left(n^{2}+1\right)(n-3)} = 0 + 0\)

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