What is the limit as x approaches 0 of (1-x/2)^{2/x} ?

The limit as x approaches 0 of (1-x/2)^{2/x} is 1/e

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limit as x approaches 0 of (1-x/2)^{2/x}

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

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

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

Apply exponent rules

= x \to 0 lim (e^{\frac{2}{x} l n (1 - \frac{x}{2})})

Apply the Limit Chain Rule:

\frac{1}{e}

= \frac{1}{e}

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

x \to - 2 - lim (- 3 ln (- 2 x - 4))

x \to 6 - lim (\frac{1}{216 - x ^{3}})

x \to 5 - lim (10 - x - x^{2})

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

What is the limit as x approaches 0 of (1-x/2)^{2/x} ?
The limit as x approaches 0 of (1-x/2)^{2/x} is 1/e