Đáp án đúng: C
Giải chi tiết:\(\mathop {\lim }\limits_{x \to 2} \frac{{\sqrt[3]{{x + 6}} - \sqrt[4]{{7x + 2}}}}{{x - 2}}\)
\(\begin{array}{l} = \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt[3]{{x + 6}} - 2 + 2 - \sqrt[4]{{7x + 2}}}}{{x - 2}}\\ = \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt[3]{{x + 6}} - 2}}{{x - 2}} - \mathop {\lim }\limits_{x \to 2} \frac{{\sqrt[4]{{7x + 2}} - 2}}{{x - 2}}\\ = \mathop {\lim }\limits_{x \to 2} \frac{{\left( {\sqrt[3]{{x + 6}} - 2} \right)\left( {{{\sqrt[3]{{x + 6}}}^2} + 2\sqrt[3]{{x + 6}} + 4} \right)}}{{\left( {x - 2} \right)\left( {{{\sqrt[3]{{x + 6}}}^2} + 2\sqrt[3]{{x + 6}} + 4} \right)}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \mathop {\lim }\limits_{x \to 2} \frac{{\left( {\sqrt[4]{{7x + 2}} - 2} \right)\left( {{{\sqrt[4]{{7x + 2}}}^3} + 2{{\sqrt[4]{{7x + 2}}}^2} + {2^2}\sqrt[4]{{7x + 2}} + {2^3}} \right)}}{{\left( {x - 2} \right)\left( {{{\sqrt[4]{{7x + 2}}}^3} + 2{{\sqrt[4]{{7x + 2}}}^2} + {2^2}\sqrt[4]{{7x + 2}} + {2^3}} \right)}}\\ = \mathop {\lim }\limits_{x \to 2} \frac{{x - 2}}{{\left( {x - 2} \right)\left( {{{\sqrt[3]{{x + 6}}}^2} + 2\sqrt[3]{{x + 6}} + 4} \right)}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \mathop {\lim }\limits_{x \to 2} \frac{{7\left( {x - 2} \right)}}{{\left( {x - 2} \right)\left( {{{\sqrt[4]{{7x + 2}}}^3} + 2{{\sqrt[4]{{7x + 2}}}^2} + {2^2}\sqrt[4]{{7x + 2}} + {2^3}} \right)}}\\ = \mathop {\lim }\limits_{x \to 2} \frac{1}{{{{\sqrt[3]{{x + 6}}}^2} + 2\sqrt[3]{{x + 6}} + 4}} - \mathop {\lim }\limits_{x \to 2} \frac{7}{{{{\sqrt[4]{{7x + 2}}}^3} + 2{{\sqrt[4]{{7x + 2}}}^2} + {2^2}\sqrt[4]{{7x + 2}} + {2^3}}}\\ = \frac{1}{{{2^2} + 2.2 + 4}} - \frac{7}{{{2^3} + {{2.2}^2} + {2^2}.2 + {2^3}}} = - \frac{{13}}{{96}}\end{array}\)
Chọn C.
