Question

Compute the limit of the sequence (n^2+1)^1/2/3n + 5n as n approaches infinity?

Answer 1

Yea

Answer 2

Sorry its 3n + 5

Answer 3

\[\lim_{n \rightarrow \infty}\frac{ \sqrt{n^2+1} }{ 3n+5 }\\
\lim_{n \rightarrow \infty}\frac{ \sqrt{n^2+1} }{3n +5}\\
\lim_{n \rightarrow \infty}\frac{ \sqrt{n^2}\sqrt{1+\frac{1}{n^2}} }{3n+5 }\\
\lim_{n \rightarrow \infty}\frac{ |n|\sqrt{1+\frac{1}{n^2}} }{3n+5 }\]
Since \(n>0\), you have \(|n|=n\):
\[\lim_{n \rightarrow \infty}\frac{ n\sqrt{1+\frac{1}{n^2}} }{3n+5 }\\
\lim_{n \rightarrow \infty}\frac{ \sqrt{1+\frac{1}{n^2}} }{3+\frac{5}{n} }\\
\]