What is the limit as x approaches 3 of (1/x)^x ?

The limit as x approaches 3 of (1/x)^x is 1/27

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limit as x approaches 3 of (1/x)^x

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

\frac{1}{27}

Decimal Notation

0.03703 \dots

Solution steps

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

Plug in the value

x = 3

= (\frac{1}{3})^{3}

Simplify

(\frac{1}{3})^{3} : \frac{1}{27}

= \frac{1}{27}

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

x \to 3 lim (\frac{( 1 + ln ( x ) )}{x})

x \to 3 lim (\frac{( 1 + x ) 2 - 1}{x})

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

x \to 3 lim (\frac{2 x + 3}{4})

What is the limit as x approaches 3 of (1/x)^x ?
The limit as x approaches 3 of (1/x)^x is 1/27