Cho hàm số \(f\left( x \right) = \frac{{\sqrt {4x + 1}  - 3}}{{x - 2}}\). Tính \(\mathop {\lim }\limits_{x \to 2} f\left( x \right).\)

A. \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) =  - \frac{2}{3}.\)

B. \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) =  - \frac{3}{2}.\)

C. \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = \frac{2}{3}.\)

D. \(\mathop {\lim }\limits_{x \to 2} f\left( x \right) = \frac{3}{2}.\)