What is the limit as x approaches infinity of 3x^3(e^{-2/(x^3)}-1) ?

The limit as x approaches infinity of 3x^3(e^{-2/(x^3)}-1) is -6

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limit as x approaches infinity of 3x^3(e^{-2/(x^3)}-1)

x \to \infty lim (3 x^{3} (e^{- \frac{2}{x ^{3}}} - 1))

- 6

Solution steps

x \to \infty lim (3 x^{3} (e^{- \frac{2}{x ^{3}}} - 1))

x \to a lim [c \cdot f (x)] = c \cdot x \to a lim f (x)

= 3 \cdot x \to \infty lim (x^{3} (e^{- \frac{2}{x ^{3}}} - 1))

Multiply by the conjugate of

e^{- \frac{2}{x ^{3}}} - 1 : \frac{e ^{- \frac{4}{x ^{3}}} - 1}{e ^{- \frac{2}{x ^{3}}} + 1}

= 3 \cdot x \to \infty lim (x^{3} \frac{e ^{- \frac{4}{x ^{3}}} - 1}{e ^{- \frac{2}{x ^{3}}} + 1})

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 3 \cdot x \to \infty lim \frac{x ^{3} ( e ^{- \frac{4}{x ^{3}}} - 1 )}{e ^{- \frac{2}{x ^{3}}} + 1}

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

= 3 \cdot \frac{lim _{x \to \infty} ( x ^{3} ( e ^{- \frac{4}{x ^{3}}} - 1 ) )}{lim _{x \to \infty} ( e ^{- \frac{2}{x ^{3}}} + 1 )}

x \to \infty lim (x^{3} (e^{- \frac{4}{x ^{3}}} - 1)) = - 4

x \to \infty lim (e^{- \frac{2}{x ^{3}}} + 1) = 2

= 3 \cdot \frac{- 4}{2}

Simplify

= - 6

\int π x cos (2 x) d x

n = 1 \sum \infty cos (\frac{1}{n ^{3}})

\int 15 x^{2} (x^{3} - 1)^{4} d x

- 15 x^{4} + 270 x^{2}

\frac{d}{d x} (5 e^{6 x})

What is the limit as x approaches infinity of 3x^3(e^{-2/(x^3)}-1) ?
The limit as x approaches infinity of 3x^3(e^{-2/(x^3)}-1) is -6