Determine the limits of the following sequences: a. a_n=(3n^{3})/(n^{3} + 1) b.

oliviayychengwh 2022-01-06 Answered
Determine the limits of the following sequences:
a. \(\displaystyle{a}_{{n}}=\frac{{{3}{n}^{{{3}}}}}{{{n}^{{{3}}}+{1}}}\)
b. \(\displaystyle{b}_{{n}}={\left(\frac{{{n}+{5}}}{{{n}}}\right)}^{{{n}}}\)
c. \(\displaystyle{c}_{{n}}={n}^{{\frac{{1}}{{n}}}}\)

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Expert Answer

Bob Huerta
Answered 2022-01-07 Author has 4348 answers
a. \(\displaystyle{a}_{{n}}={\frac{{{3}{n}^{{{3}}}}}{{{n}^{{{3}}}+{1}}}}\)
\(\displaystyle\lim_{{{n}\rightarrow\infty}}{a}_{{n}}=\lim_{{{n}\rightarrow\infty}}{\frac{{{3}{n}^{{{3}}}}}{{{n}^{{{2}}}+{1}}}}\)
\(\displaystyle=\lim_{{{n}\rightarrow\infty}}{\frac{{{3}}}{{{1}+{\frac{{{1}}}{{{n}^{{{3}}}}}}}}}\)
\(\displaystyle={\frac{{{3}}}{{{1}+{0}}}}\)
\(\displaystyle={3}\)
b.\(\displaystyle{b}_{{n}}={\left({\frac{{{n}+{5}}}{{{n}}}}\right)}^{{{n}}}\)
\(\displaystyle\lim_{{{n}\rightarrow\infty}}{b}_{{n}}=\lim_{{{n}\rightarrow\infty}}{\left({\frac{{{n}+{5}}}{{{n}}}}\right)}^{{{n}}}\)
\(\displaystyle=\lim_{{{n}\rightarrow\infty}}{\left({1}-{\frac{{{5}}}{{{n}}}}\right)}^{{{n}}}\)
\(\displaystyle{e}^{{{5}}}\)
c. \(\displaystyle{c}_{{n}}={n}^{{{\frac{{{1}}}{{{n}}}}}}\)
\(\displaystyle\lim_{{{n}\rightarrow\infty}}{n}^{{{\frac{{{1}}}{{{n}}}}}}={1}\)
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