limit as x approaches 0 of (\sqrt[3]{x^3+8}-sqrt(x^2+4))/(x^2)

解答

x \to 0 lim (\frac{3 x ^{3} + 8 - x ^{2} + 4}{x ^{2}})

- \frac{1}{4}

十进制

- 0.25

求解步骤

x \to 0 lim (\frac{3 x ^{3} + 8 - x ^{2} + 4}{x ^{2}})

使用洛必达法则

= x \to 0 lim \frac{\frac{x ^{2}}{( x ^{3} + 8 ) ^{\frac{2}{3}}} - \frac{x}{x ^{2} + 4}}{2 x}

化简

\frac{\frac{x ^{2}}{( x ^{3} + 8 ) ^{\frac{2}{3}}} - \frac{x}{x ^{2} + 4}}{2 x} : \frac{x x ^{2} + 4 - ( x ^{3} + 8 ) ^{\frac{2}{3}}}{2 ( x ^{3} + 8 ) ^{\frac{2}{3}} x ^{2} + 4}

= x \to 0 lim \frac{x x ^{2} + 4 - ( x ^{3} + 8 ) ^{\frac{2}{3}}}{2 ( x ^{3} + 8 ) ^{\frac{2}{3}} x ^{2} + 4}

代入值

x = 0

= \frac{0 \cdot 0 ^{2} + 4 - ( 0 ^{3} + 8 ) ^{\frac{2}{3}}}{2 ( 0 ^{3} + 8 ) ^{\frac{2}{3}} 0 ^{2} + 4}

化简

\frac{0 \cdot 0 ^{2} + 4 - ( 0 ^{3} + 8 ) ^{\frac{2}{3}}}{2 ( 0 ^{3} + 8 ) ^{\frac{2}{3}} 0 ^{2} + 4} : - \frac{1}{4}

= - \frac{1}{4}

x \to 0 lim (\frac{1}{x + 1})

x \to 0 lim ((\frac{a ^{x} + b ^{x}}{2})^{\frac{1}{x}})

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x \to 0 + lim ((tan (x))^{s i n (x)})