What is the limit as x approaches 0 of (1+3x)^{-5/x} ?

The limit as x approaches 0 of (1+3x)^{-5/x} is 1/(e^{15)}

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limit as x approaches 0 of (1+3x)^{-5/x}

x \to 0 lim ((1 + 3 x)^{- \frac{5}{x}})

\frac{1}{e ^{15}}

Decimal Notation

3.05902 E - 7

Solution steps

x \to 0 lim ((1 + 3 x)^{- \frac{5}{x}})

Simplify

(1 + 3 x)^{- \frac{5}{x}} : \frac{1}{( 1 + 3 x ) ^{\frac{5}{x}}}

= x \to 0 lim (\frac{1}{( 1 + 3 x ) ^{\frac{5}{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 + 3 x ) ^{\frac{5}{x}} )}

x \to 0 lim (1) = 1

x \to 0 lim ((1 + 3 x)^{\frac{5}{x}}) = e^{15}

= \frac{1}{e ^{15}}

x \to 5 - lim (arcsin (\frac{x}{5}))

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

x \to 2 - lim ((x - 2) ln (2 - x))

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

x \to - \infty lim (2 \cdot (x))

What is the limit as x approaches 0 of (1+3x)^{-5/x} ?
The limit as x approaches 0 of (1+3x)^{-5/x} is 1/(e^{15)}