What is the limit as x approaches infinity of ((x-3)/(x-4))^{-x} ?

The limit as x approaches infinity of ((x-3)/(x-4))^{-x} is 1/e

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limit as x approaches infinity of ((x-3)/(x-4))^{-x}

x \to \infty lim ((\frac{x - 3}{x - 4})^{- x})

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

x \to \infty lim ((\frac{x - 3}{x - 4})^{- x})

Apply exponent rules

= x \to \infty lim (e^{- x l n (\frac{x - 3}{x - 4})})

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

What is the limit as x approaches infinity of ((x-3)/(x-4))^{-x} ?
The limit as x approaches infinity of ((x-3)/(x-4))^{-x} is 1/e