What is the limit as x approaches infinity of 4x^3(e^{-2/(x^3)}-1) ?

The limit as x approaches infinity of 4x^3(e^{-2/(x^3)}-1) is -8

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limit as x approaches infinity of 4x^3(e^{-2/(x^3)}-1)

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

- 8

Solution steps

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

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

= 4 \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}

= 4 \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}

= 4 \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

= 4 \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

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

Simplify

4 \cdot \frac{- 4}{2} : - 8

= - 8

\int sin^{4} (d) x d x

x = y^{2} + 6, y = \frac{x}{2} - \frac{3}{2}

t \to 0 lim (cos (t) sin (t))

\int cos^{5} (x) sin^{2} (x) d x

2 y^{3} + x y - y = 250 x^{4}, (1, 5)

What is the limit as x approaches infinity of 4x^3(e^{-2/(x^3)}-1) ?
The limit as x approaches infinity of 4x^3(e^{-2/(x^3)}-1) is -8