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$\lim _{n \rightarrow \infty} \frac{a^{n}+b^{n}}{a^{n}-b^{n}},$ where $a>b>1,$ is equal to
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limit $=\lim _{n \rightarrow \infty} \frac{1+\left(\frac{b}{a}\right)^{n}}{1-\left(\frac{b}{a}\right)^{n}}=1,$
because $0<\frac{b}{a}<1$ implies
$\left(\frac{b}{a}\right)^{n} \rightarrow 0$ as $n \rightarrow \infty$
because $0<\frac{b}{a}<1$ implies
$\left(\frac{b}{a}\right)^{n} \rightarrow 0$ as $n \rightarrow \infty$
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