Cho Lim căn 3x + 4} - 4 / x - 4 = a/b

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Cho \[\mathop {\lim }\limits_{x \to 4} \frac{{\sqrt {3x + 4} - 4}}{{x - 4}} = \frac{a}{b}\], với \[\frac{a}{b}\] là phân số tối giản. Tính \[2a + {b^2}\]?

\[14\].

\[66\].

\[22\].

\[70\].

Giải thích

Chọn D.

Ta có: \[\begin{array}{l}\mathop {\lim }\limits_{x \to 4} \frac{{\sqrt {3x + 4} - 4}}{{x - 4}} = \mathop {\lim }\limits_{x \to 4} \frac{{\left( {\sqrt {3x + 4} - 4} \right)\left( {\sqrt {3x + 4} + 4} \right)}}{{\left( {x - 4} \right)\left( {\sqrt {3x + 4} + 4} \right)}} = \mathop {\lim }\limits_{x \to 4} \frac{{3x - 12}}{{\left( {x - 4} \right)\left( {\sqrt {3x + 4} + 4} \right)}}\\ = \mathop {\lim }\limits_{x \to 4} \frac{{3\left( {x - 4} \right)}}{{\left( {x - 4} \right)\left( {\sqrt {3x + 4} + 4} \right)}} = \mathop {\lim }\limits_{x \to 4} \frac{3}{{\sqrt {3x + 4} + 4}} = \frac{3}{{\sqrt {3.4 + 4} + 4}} = \frac{3}{8} = \frac{a}{b} \Rightarrow \left\{ {\begin{array}{*{20}{c}}{a = 3}\\{b = 8}\end{array}} \right..\end{array}\]

Vậy \[2a + {b^2} = 2.3 + {8^2} = 70.\]