Cho đa thức \(f\left( x \right)\) thỏa mãn Tính \(\mathop {\lim }\limits_{x \to 3} \frac{{\sqrt[3]{{5f\left( x \right) - 11}} - 4}}{{{x^2} - x - 6}}.\) Đáp án: ……….

Đặt\(\mathop {\lim }\limits_{x \to 3} \frac{{f\left( x \right) - 15}}{{x - 3}} = g\left( x \right) \Rightarrow f\left( x \right) = \left( {x - 3} \right)g\left( x \right) + 15 \Rightarrow \mathop {\lim f\left( x \right) = 15}\limits_{x \to 3} \).

Khi đó \(\mathop {\lim }\limits_{x \to 3} \frac{{\sqrt[3]{{5f\left( x \right) - 11}} - 4}}{{{x^2} - x - 6}} = \mathop {\lim }\limits_{x \to 3} \frac{{5f\left( x \right) - 11 - 64}}{{{{\left( {\sqrt[3]{{5f\left( x \right) - 11}}} \right)}^2} + 4\sqrt[3]{{5f\left( x \right) - 11}} + 16}} \cdot \frac{1}{{\left( {x - 3} \right)\left( {x + 2} \right)}}\)

 \( = \mathop {\lim }\limits_{x \to 3} \frac{{5\left[ {f\left( x \right) - 15} \right]}}{{x - 3}} \cdot \frac{1}{{\left( {x + 2} \right)\left[ {{{\left( {\sqrt[3]{{5f\left( x \right) - 11}}} \right)}^2} + 4\sqrt[3]{{5f\left( x \right) - 11}} + 16} \right]}}\)\( = 5 \cdot 12 \cdot \frac{1}{{5 \cdot \left[ {16 + 16 + 16} \right]}} = \frac{1}{4}.\)

Đáp án: \(\frac{1}{4}.\)