What is the limit as x approaches 0 of (1+x)^{-3/x} ?

The limit as x approaches 0 of (1+x)^{-3/x} is 1/(e^3)

Popular >

limit as x approaches 0 of (1+x)^{-3/x}

x \to 0 lim ((1 + x)^{- \frac{3}{x}})

\frac{1}{e ^{3}}

Decimal

0.04978 \dots

Solution steps

x \to 0 lim ((1 + x)^{- \frac{3}{x}})

Simplify

(1 + x)^{- \frac{3}{x}} : \frac{1}{( 1 + x ) ^{\frac{3}{x}}}

= x \to 0 lim (\frac{1}{( 1 + x ) ^{\frac{3}{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{3}{x}} )}

x \to 0 lim (1) = 1

x \to 0 lim ((1 + x)^{\frac{3}{x}}) = e^{3}

= \frac{1}{e ^{3}}

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

x \to - \infty lim (- 1 x)

y \to - 8 lim ((19 - y)^{\frac{2}{3}})

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

x \to 0 + lim (x (ln (x))^{6})

What is the limit as x approaches 0 of (1+x)^{-3/x} ?
The limit as x approaches 0 of (1+x)^{-3/x} is 1/(e^3)