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

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

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

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

e^{3}

Decimal Notation

20.08553 \dots

Solution steps

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

Apply exponent rules

= x \to \infty lim (e^{3 x l n (\frac{x + 2}{x + 1})})

Apply the Limit Chain Rule:

e^{3}

= e^{3}

x \to 2 lim (5 - x)

r \to 0 lim (1 + \frac{1}{r} + \frac{1}{r ^{2}})

x \to 0 lim (\frac{1 0 ^{x} - 5 ^{x}}{x})

x \to - \infty lim (5 x - 1)

x \to 0 lim (\frac{x}{e ^{\frac{1}{x}}})

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