# Compute the limit of the sequences: (a)(1-\frac{3}{n})^{2n} (b)3^{1/n}

Compute the limit of the sequences:
$$\displaystyle{\left({a}\right)}{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}^{{{2}{n}}}$$
$$\displaystyle{\left({b}\right)}{3}^{{\frac{{1}}{{n}}}}$$

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(a)
$$\displaystyle{\left({a}\right)}{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}^{{{2}{n}}}$$
$$\displaystyle\lim_{{{n}\rightarrow{d}}}{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}^{{{2}{n}}}$$
$$\displaystyle=\lim_{{{n}\rightarrow{d}}}{e}^{{{\ln{{\left[{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}^{{{2}{n}}}\right]}}}}}$$
$$\displaystyle=\lim_{{{n}\rightarrow{d}}}{e}^{{{2}{n}\times{\ln{{\left({1}-{\frac{{{3}}}{{{n}}}}\right\rbrace}}}}}$$
$$\displaystyle={e}^{{\lim_{{{n}\rightarrow{d}}}{2}{n}\times{\ln{{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}}}}}$$
$$\displaystyle={e}^{{{2}\lim_{{{n}\rightarrow{d}}}{n}\times{\ln{{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}}}}}$$
$$\displaystyle={e}^{{\lim_{{{n}\rightarrow{d}}}}}{\frac{{{\ln{{\left({1}-{\frac{{{3}}}{{{n}}}}\right)}}}}}{{\frac{{1}}{{n}}}}}$$
$$\displaystyle{e}^{{\lim_{{{n}\rightarrow{d}}}}}{\left({\frac{{{\frac{{-{1}}}{{{1}-{\frac{{{3}}}{{{n}}}}}}}{x}}}{{\frac{{1}}{{n}^{{{2}}}}}}}\right)}$$
$$\displaystyle{e}^{{2}}{x}{\left\lbrace\lim_{{{n}\rightarrow{d}}}\right\rbrace}{\frac{{-{3}}}{{{1}-\frac{{3}}{{n}}}}}$$
$$\displaystyle{e}^{{{2}{x}{\left(-{3}\right)}\lim_{{{n}\rightarrow{d}}}}}{\frac{{{1}}}{{{1}-\frac{{3}}{{n}}}}}$$
$$\displaystyle{e}^{{-{6}{x}}}{\frac{{{1}}}{{{1}-\frac{{3}}{{n}}}}}={e}^{{-{6}{x}{\frac{{{1}}}{{{1}-{0}}}}}}$$
$$\displaystyle{e}^{{-{6}\times{1}}}={e}^{{-{6}}}$$
(b)
$$\displaystyle{3}^{{\frac{{1}}{{n}}}}$$
$$\displaystyle\lim_{{{n}\rightarrow{d}}}{3}^{{\frac{{1}}{{n}}}}$$
$$\displaystyle{3}^{{{0}}}={1}$$