Cho a là hằng số dương, lim x → a a ( √ 3 x 2 + a 2 − 2 a ) x − a bằng

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Cho \(a\) là hằng số dương, \(\mathop {\lim }\limits_{x \to a} \frac{{a\left( {\sqrt {3{x^2} + {a^2}} - 2a} \right)}}{{x - a}}\) bằng

\(\frac{{3a}}{2}\).

\(\frac{3}{2}\).

\(\frac{{3{a^2}}}{2}\).

\(\frac{{2a}}{3}\).

Giải thích

Ta có \(\mathop {\lim }\limits_{x \to a} \frac{{a\left( {\sqrt {3{x^2} + {a^2}} - 2a} \right)}}{{x - a}} = \mathop {\lim }\limits_{x \to a} \frac{{a\left( {\sqrt {3{x^2} + {a^2}} - 2a} \right)\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}}{{(x - a)\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}}\)

\( = \mathop {\lim }\limits_{x \to a} \frac{{a\left( {3{x^2} + {a^2} - 4{a^2}} \right)}}{{\left( {x - a} \right)\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}} = \mathop {\lim }\limits_{x \to a} \frac{{a\left( {3{x^2} - 3{a^2}} \right)}}{{(x - a)\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}}\)

\( = \mathop {\lim }\limits_{x \to a} \frac{{3a\left( {x - a} \right)\left( {x + a} \right)}}{{(x - a)\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}} = = \mathop {\lim }\limits_{x \to a} \frac{{3a\left( {x + a} \right)}}{{\left( {\sqrt {3{x^2} + {a^2}} + 2a} \right)}} = \frac{{3a\left( {a + a} \right)}}{{\sqrt {3{a^2} + {a^2}} + 2a}} = \frac{{6{a^2}}}{{4a}} = \frac{{3a}}{2}\). Chọn A.