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

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

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

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

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

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

Simplify

(1 - \frac{1}{x})^{x - 1} : (1 - \frac{1}{x})^{x} (1 - \frac{1}{x})^{- 1}

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

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

With the exception of indeterminate form

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

x \to \infty lim ((1 - \frac{1}{x})^{x}) = \frac{1}{e}

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

= \frac{1}{e} \cdot 1

Simplify

= \frac{1}{e}

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