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

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

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

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

\frac{1}{e}

Decimal Notation

0.36787 \dots

Solution steps

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

Simplify

(1 + x)^{- \frac{1}{x}} : \frac{1}{( 1 + x ) ^{\frac{1}{x}}}

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

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to 0} ( 1 )}{lim _{x \to 0} ( ( 1 + x ) ^{\frac{1}{x}} )}

x \to 0 lim (1) = 1

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

= \frac{1}{e}

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\int_{0}^{1} 4 x d x

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