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

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

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

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

\frac{1}{e}

Decimal

0.36787 \dots

Solution steps

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

x \to a - lim f (x) = x \to a + lim f (x) = L

then

x \to a lim f (x) = L

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

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

= \frac{1}{e}

x \to 2 lim (\frac{2 x - 5}{x - 2})

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

x \to \infty lim (\frac{24}{x})

y \to - 2 lim (\frac{y ^{3} - 8}{2 y ^{2} - 7 y + 12})

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

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