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

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

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

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

\frac{1}{e ^{2}}

Decimal Notation

0.13533 \dots

Solution steps

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

Simplify

(1 + x)^{- \frac{2}{x}} : \frac{1}{( 1 + x ) ^{\frac{2}{x}}}

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

x \to 0 lim (1) = 1

x \to 0 lim ((1 + x)^{\frac{2}{x}}) = e^{2}

= \frac{1}{e ^{2}}

x \to 0 lim (cos^{\frac{3}{x}} (2 x))

x \to - 3 - lim (\frac{3 x}{9 - x ^{2}})

x \to 0 + lim ((x^{2} + 1) ln (x))

x \to 6 lim (3 x - 4)

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

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