Đáp án:
\[\mathop {\lim }\limits_{x \to {3^ + }} \frac{{x - 3}}{{\sqrt[3]{x} - \sqrt[3]{3}}} = 3.\sqrt[3]{9}\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\mathop {\lim }\limits_{x \to {3^ + }} \frac{{x - 3}}{{\sqrt[3]{x} - \sqrt[3]{3}}}\\
= \mathop {\lim }\limits_{x \to {3^ + }} \frac{{{{\left( {\sqrt[3]{x}} \right)}^3} - {{\left( {\sqrt[3]{3}} \right)}^3}}}{{\sqrt[3]{x} - \sqrt[3]{3}}}\\
= \mathop {\lim }\limits_{x \to {3^ + }} \frac{{\left( {\sqrt[3]{x} - \sqrt[3]{3}} \right).\left( {{{\sqrt[3]{x}}^2} + \sqrt[3]{x}.\sqrt[3]{3} + {{\sqrt[3]{3}}^2}} \right)}}{{\sqrt[3]{x} - \sqrt[3]{3}}}\\
= \mathop {\lim }\limits_{x \to {3^ + }} \left( {\sqrt[3]{{{x^2}}} + \sqrt[3]{{3x}} + \sqrt[3]{9}} \right)\\
= \sqrt[3]{{{3^2}}} + \sqrt[3]{{3.3}} + \sqrt[3]{9}\\
= 3.\sqrt[3]{9}
\end{array}\)
