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

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

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

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

0

Solution steps

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

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

x \to 0 - lim (x) = 0

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

= 0 \cdot \frac{1}{e}

Simplify

= 0

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