Phương pháp giải:

Sử dụng phương pháp nhân với biểu thức liên hợp để khử dạng \(\dfrac{0}{0}\).

Lời giải chi tiết:

\(\begin{array}{l}\mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}}\sqrt {3x + 1}  - 4}}{{x - 1}}\\ = \mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}}\left( {\sqrt {3x + 1}  - 2} \right) + 2\left( {\sqrt[3]{{7x + 1}} - 2} \right)}}{{x - 1}}\\ = \mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}}\left( {\sqrt {3x + 1}  - 2} \right)}}{{x - 1}} + 2\mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}} - 2}}{{x - 1}}\\ = \mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}}\left( {\sqrt {3x + 1}  - 2} \right)\left( {\sqrt {3x + 1}  + 2} \right)}}{{\left( {x - 1} \right)\left( {\sqrt {3x + 1}  + 2} \right)}} + 2\mathop {\lim }\limits_{x \to 1} \dfrac{{\left( {\sqrt[3]{{7x + 1}} - 2} \right)\left( {{{\sqrt[3]{{7x + 1}}}^2} + 2\sqrt[3]{{7x + 1}} + 4} \right)}}{{\left( {x - 1} \right)\left( {{{\sqrt[3]{{7x + 1}}}^2} + 2\sqrt[3]{{7x + 1}} + 4} \right)}}\\ = \mathop {\lim }\limits_{x \to 1} \dfrac{{\sqrt[3]{{7x + 1}}\left( {3x + 1 - 4} \right)}}{{\left( {x - 1} \right)\left( {\sqrt {3x + 1}  + 2} \right)}} + 2\mathop {\lim }\limits_{x \to 1} \dfrac{{7x + 1 - 8}}{{\left( {x - 1} \right)\left( {{{\sqrt[3]{{7x + 1}}}^2} + 2\sqrt[3]{{7x + 1}} + 4} \right)}}\\ = \mathop {\lim }\limits_{x \to 1} \dfrac{{3\sqrt[3]{{7x + 1}}}}{{\sqrt {3x + 1}  + 2}} + 2\mathop {\lim }\limits_{x \to 1} \dfrac{7}{{{{\sqrt[3]{{7x + 1}}}^2} + 2\sqrt[3]{{7x + 1}} + 4}}\\ = \dfrac{{3.2}}{{2 + 2}} + 2.\dfrac{7}{{{2^2} + 2.2 + 4}} = \dfrac{3}{2} + \dfrac{7}{6} = \dfrac{8}{3}\end{array}\)

Chọn B.