lim_(n rarr oo)((a-1+root(n)(b))/(a))^(n)

Rewrite Limit Expression: First, let's rewrite the limit expression to make it clearer. lim_(n rarr oo)((a-1+root(n)(b))/a)^(n)Rewrite Inside Parentheses: Notice that the expression inside the parentheses can be rewritten as 1 + (root(n)(b) - 1)/a. So, we have lim_(n rarr oo)(1 + (root(n)(b) - 1)/a)^(n)Use Limit Property: Now, let's use the fact that lim_(n rarr oo)(1 + x/n)^n = e^x when n approaches infinity. Here, x = (root(n)(b) - 1)/a.Determine x Value: As n approaches infinity, root(n)(b) approaches 0, so (root(n)(b) - 1)/a approaches -1/a.Calculate Final Limit: Therefore, the limit becomes e^(-1/a).Final Answer: So, the final answer is e^(-1/a).

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lim_(n rarr oo)((a-1+root(n)(b))/(a))^(n)

$\lim _{n \rightarrow \infty}\left(\frac{a-1+\sqrt[n]{b}}{a}\right)^{n}$

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Precalculus

Evaluate rational exponents

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\lim _{n \rightarrow \infty}\left(\frac{a-1+\sqrt[n]{b}}{a}\right)^{n}

Rewrite Limit Expression: First, let's rewrite the limit expression to make it clearer. $\lim_{n \rightarrow \infty}\left(\frac{a-1+\sqrt[n]{b}}{a}\right)^{n}$
Rewrite Inside Parentheses: Notice that the expression inside the parentheses can be rewritten as $1 + (\sqrt[n]{b} - 1)/a$ . So, we have $\lim_{n \to \infty}(1 + (\sqrt[n]{b} - 1)/a)^{n}$
Use Limit Property: Now, let's use the fact that $\lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n = e^x$ when $n$ approaches infinity. $\newline$ Here, $x = \frac{\sqrt[n]{b} - 1}{a}$ .
Determine $x$ Value: As $n$ approaches infinity, $\sqrt[n]{b}$ approaches $0$ , so $(\sqrt[n]{b} - 1)/a$ approaches $-1/a$ .
Calculate Final Limit: Therefore, the limit becomes $e^{(-1/a)}$ .
Final Answer: So, the final answer is $e^{(-1/a)}$ .