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

解答

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

- 8

求解步骤

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))

乘以

e^{- \frac{2}{x ^{3}}} - 1

的共轭根式

: \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})

分式相乘:

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

不定型除外

= 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}

化简

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

= - 8

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