What is the limit as x approaches infinity of (8^x)/(9^{x+1)} ?

The limit as x approaches infinity of (8^x)/(9^{x+1)} is 0

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limit as x approaches infinity of (8^x)/(9^{x+1)}

x \to \infty lim (\frac{8 ^{x}}{9 ^{x + 1}})

0

Solution steps

x \to \infty lim (\frac{8 ^{x}}{9 ^{x + 1}})

Simplify

9^{x + 1} : 9^{x} \cdot 9

= x \to \infty lim (\frac{8 ^{x}}{9 ^{x} \cdot 9})

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

= \frac{1}{9} \cdot x \to \infty lim (\frac{8 ^{x}}{9 ^{x}})

Apply exponent rule:

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

= \frac{1}{9} \cdot x \to \infty lim ((\frac{8}{9})^{x})

Apply the common limit:

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

= \frac{1}{9} \cdot 0

Simplify

= 0

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What is the limit as x approaches infinity of (8^x)/(9^{x+1)} ?
The limit as x approaches infinity of (8^x)/(9^{x+1)} is 0