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

The limit as x approaches infinity of (2^{x+1})/(3^{x-1)} is 0

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

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

0

Solution steps

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

Simplify

2^{x + 1} : 2^{x} \cdot 2

= x \to \infty lim (\frac{2 ^{x} \cdot 2}{3 ^{x - 1}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 2 \cdot x \to \infty lim (\frac{2 ^{x}}{3 ^{x - 1}})

Simplify

3^{x - 1} : 3^{x} \cdot 3^{- 1}

= 2 \cdot x \to \infty lim (\frac{2 ^{x}}{3 ^{x} \cdot 3 ^{- 1}})

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 2 \cdot \frac{1}{3 ^{- 1}} \cdot x \to \infty lim (\frac{2 ^{x}}{3 ^{x}})

\frac{1}{3 ^{- 1}} = 3

= 2 \cdot 3 \cdot x \to \infty lim (\frac{2 ^{x}}{3 ^{x}})

Apply exponent rule:

\frac{a ^{c}}{b ^{c}} = (\frac{a}{b})^{c}

= 2 \cdot 3 \cdot x \to \infty lim ((\frac{2}{3})^{x})

Apply the common limit:

x \to \infty lim (a^{x}) = 0, 0 < a < 1

= 2 \cdot 3 \cdot 0

Simplify

= 0

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What is the limit as x approaches infinity of (2^{x+1})/(3^{x-1)} ?
The limit as x approaches infinity of (2^{x+1})/(3^{x-1)} is 0