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limit as x approaches 0+of xln^2(x)

解答

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

解答

0

求解步骤

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

改写以应用洛必达法则

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

使用洛必达法则

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

化简

\frac{\frac{2 l n ( x )}{x}}{- \frac{1}{x ^{2}}} : - 2 x ln (x)

= x \to 0 + lim (- 2 x ln (x))

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

= - 2 \cdot x \to 0 + lim (x ln (x))

改写以应用洛必达法则

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

使用洛必达法则

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

化简

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

= - 2 \cdot x \to 0 + lim (- x)

代入值

x = 0

= - 2 (- 0)

化简

= 0

作图

流行的例子

limit as x approaches 0-of (1-e^x)^{x^2}

x \to 0 - lim ((1 - e^{x})^{x^{2}})

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

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

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

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

limit as x approaches 0 of ln(1-e^{-x})

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

limit as x approaches 5 of x^2-4x+3

x \to 5 lim (x^{2} - 4 x + 3)