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

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

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

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

\frac{1}{e}

Decimal

0.60653 \dots

Solution steps

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

Apply exponent rule:

a^{- 1} = \frac{1}{a}

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

Apply exponent rules

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

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

x \to 0 lim (\frac{x ^{2} + 4 x - 5 - 5}{x})

x \to 1 lim (x^{c s c (π) x})

x \to 0 lim ((x^{2}) cos (\frac{1}{x}))

u \to 0 lim (cot (u) - csc (u))

x \to 0 lim ((x)^{x^{2}})

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