Đáp án:
\[\mathop {\lim }\limits_{x \to + \infty } \left( {3x + 2 - \sqrt {9{x^2} + 12x - 3} } \right) = 0\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\mathop {\lim }\limits_{x \to + \infty } \left( {3x + 2 - \sqrt {9{x^2} + 12x - 3} } \right)\\
= \mathop {\lim }\limits_{x \to + \infty } \frac{{{{\left( {3x + 2} \right)}^2} - {{\sqrt {9{x^2} + 12x - 3} }^2}}}{{3x + 2 + \sqrt {9{x^2} + 12x - 3} }}\\
= \mathop {\lim }\limits_{x \to + \infty } \frac{{9{x^2} + 12x + 4 - 9{x^2} - 12x + 3}}{{3x + 2 + \sqrt {9{x^2} + 12x - 3} }}\\
= \mathop {\lim }\limits_{x \to + \infty } \frac{7}{{3x + 2 + \sqrt {9{x^2} + 12x - 3} }}\\
\mathop {\lim }\limits_{x \to + \infty } \left( {3x + 2 + \sqrt {9{x^2} + 12x - 3} } \right) = + \infty \\
\Rightarrow \mathop {\lim }\limits_{x \to + \infty } \frac{7}{{3x + 2 + \sqrt {9{x^2} + 12x - 3} }} = 0\\
\Rightarrow \mathop {\lim }\limits_{x \to + \infty } \left( {3x + 2 - \sqrt {9{x^2} + 12x - 3} } \right) = 0
\end{array}\)
