What is the limit as n approaches infinity of ((3^n+2)/(3^n-1))^{3^n} ?

The limit as n approaches infinity of ((3^n+2)/(3^n-1))^{3^n} is e^3

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limit as n approaches infinity of ((3^n+2)/(3^n-1))^{3^n}

n \to \infty lim ((\frac{3 ^{n} + 2}{3 ^{n} - 1})^{3^{n}})

e^{3}

Decimal

20.08553 \dots

Solution steps

n \to \infty lim ((\frac{3 ^{n} + 2}{3 ^{n} - 1})^{3^{n}})

Apply the Limit Chain Rule:

e^{3}

= e^{3}

x \to 0 + lim \frac{( 1 - 1 0 ^{- x^{2}} )}{x ^{2} ln ( 10 )}

x \to 0 lim (\frac{x}{tan ( 4 x )})

x \to 0 lim (\frac{2 x ^{2}}{3 - 3 cos ( x )})

x \to 1 lim (\frac{1}{5 - x})

x \to 0 lim ((x + e^{x})^{\frac{2}{x}})

What is the limit as n approaches infinity of ((3^n+2)/(3^n-1))^{3^n} ?
The limit as n approaches infinity of ((3^n+2)/(3^n-1))^{3^n} is e^3