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

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

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

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

\frac{1}{e ^{2}}

Decimal

0.13533 \dots

Solution steps

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

Simplify

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

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

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

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

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

= \frac{1}{e ^{2}} \cdot 1

Simplify

= \frac{1}{e ^{2}}

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