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

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

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

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

\frac{1}{e}

Decimal Notation

0.36787 \dots

Solution steps

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

Apply exponent rules

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

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

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

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

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

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

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

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