limit as x approaches 0+of x^2ln(x^3)

解答

x \to 0 + lim (x^{2} ln (x^{3}))

0

求解步骤

x \to 0 + lim (x^{2} ln (x^{3}))

ln (x^{3}) = 3 ln (x)

= x \to 0 + lim (x^{2} \cdot 3 ln (x))

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

= 3 \cdot x \to 0 + lim (x^{2} ln (x))

改写以应用洛必达法则

= 3 \cdot x \to 0 + lim (\frac{ln ( x )}{\frac{1}{x ^{2}}})

使用洛必达法则

= 3 \cdot x \to 0 + lim (\frac{\frac{1}{x}}{- \frac{2}{x ^{3}}})

化简

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

= 3 \cdot x \to 0 + lim (- \frac{x ^{2}}{2})

代入值

x = 0

= 3 (- \frac{0 ^{2}}{2})

化简

3 (- \frac{0 ^{2}}{2}) : 0

= 0

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x \to \infty lim (- \frac{x}{0})

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