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

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

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

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

0

Solution steps

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

Simplify

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

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

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

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

Simplify

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

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

Simplify

2^{x} \cdot \frac{1}{3 ^{x}} : (\frac{2}{3})^{x}

= 2 \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 0

Simplify

= 0

x \to 0 lim (x - arcsin (x))

x \to - 10 - lim (\frac{1}{( x + 10 ) ^{6}})

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

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

x \to 0 + lim ((e^{x} - 1) cot (x))

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