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

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

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

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

\frac{1}{e ^{3}}

Decimal

0.04978 \dots

Solution steps

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

Simplify

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

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

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

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

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

= \frac{1}{e ^{3}} \cdot 1

Simplify

= \frac{1}{e ^{3}}

x \to 0 + lim (x^{4} sin (\frac{1}{x}))

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

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

y \to 0 lim (y + y^{3} sin (\frac{x}{y}))

x \to 0 lim (∣ cos (x) ∣^{\frac{1}{x}})

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