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

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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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. $$\displaystyle{c}_{{n}}={n}^{{{\frac{{{1}}}{{{n}}}}}}$$
$$\displaystyle\lim_{{{n}\rightarrow\infty}}{n}^{{{\frac{{{1}}}{{{n}}}}}}={1}$$