Đáp án đúng: D
Giải chi tiết:Đặt \(\dfrac{{f\left( x \right) - 2}}{{x - 4}} = g\left( x \right) \Rightarrow f\left( x \right) = \left( {x - 4} \right)g\left( x \right) + 2\)
\( \Rightarrow \mathop {\lim }\limits_{x \to 4} f\left( x \right) = 2\).
\(\begin{array}{l}L = \mathop {\lim }\limits_{x \to 4} \dfrac{{\sqrt[3]{{3f\left( x \right) + 21}} + \sqrt[4]{{4f\left( x \right) + 8}} - 5}}{{{x^2} - 16}}\\\,\,\,\, = \mathop {\lim }\limits_{x \to 4} \dfrac{{\sqrt[3]{{3f\left( x \right) + 21}} - 3}}{{{x^2} - 16}} + \mathop {\lim }\limits_{x \to 4} \dfrac{{\sqrt[4]{{4f\left( x \right) + 8}} - 2}}{{{x^2} - 16}}\\\,\,\,\, = \mathop {\lim }\limits_{x \to 4} \dfrac{{3f\left( x \right) + 21 - 27}}{{\left( {{x^2} - 16} \right)\left[ {{{\sqrt[3]{{3f\left( x \right) + 21}}}^2} + 3\sqrt[3]{{3f\left( x \right) + 21}} + 9} \right]}}\\ + \mathop {\lim }\limits_{x \to 4} \dfrac{{4f\left( x \right) + 8 - 16}}{{\left( {{x^2} - 16} \right)\left[ {{{\sqrt[4]{{4f\left( x \right) + 8}}}^3} + 2{{\sqrt[4]{{4f\left( x \right) + 8}}}^2} + 4\sqrt[4]{{4f\left( x \right) + 8}} + 8} \right]}}\\\,\,\,\, = 3.\mathop {\lim }\limits_{x \to 4} \dfrac{{f\left( x \right) - 2}}{{x - 4}}.\dfrac{1}{{\left( {x + 4} \right)\left[ {{{\sqrt[3]{{3f\left( x \right) + 21}}}^2} + 3\sqrt[3]{{3f\left( x \right) + 21}} + 9} \right]}}\\ + 4\mathop {\lim }\limits_{x \to 4} \dfrac{{f\left( x \right) - 2}}{{x - 4}}.\dfrac{1}{{\left( {x + 4} \right)\left[ {{{\sqrt[4]{{4f\left( x \right) + 8}}}^3} + 2{{\sqrt[4]{{4f\left( x \right) + 8}}}^2} + 4\sqrt[4]{{4f\left( x \right) + 8}} + 8} \right]}}\\\,\,\,\, = 3.4.\dfrac{1}{{8.\left( {9 + 9 + 9} \right)}} + 4.4.\dfrac{1}{{8\left( {8 + 8 + 8 + 8} \right)}} = \dfrac{{17}}{{144}}\\ \Rightarrow a = 17,\,\,b = 144\\ \Rightarrow b - 5a - 35 = 24.\end{array}\)
Chọn D.