\(\lim\limits_{x\rightarrow2}\dfrac{\sqrt{4x^2-6x}-x-2}{x^2+x-6}\)
\(=\lim\limits_{x\rightarrow2}\left(\dfrac{4x^2-6x-x^2-4x-4}{\sqrt{4x^2-6x}+x+2}\cdot\dfrac{1}{\left(x+3\right)\left(x-2\right)}\right)\)
\(=\lim\limits_{x\rightarrow2}\left(\dfrac{3x^2-10x-4}{\left(x+3\right)\left(x-2\right)\cdot\left(\sqrt{4x^2-6x}+x+2\right)}\right)\)
\(=\lim\limits_{x\rightarrow2}\left(\dfrac{\left(x-2\right)\left(3x+2\right)}{\left(x+3\right)\left(x-2\right)\cdot\left(\sqrt{4x^2-6x}+x+2\right)}\right)\)
\(=\lim\limits_{x\rightarrow2}\dfrac{3x+2}{\left(x+3\right)\left(\sqrt{4x^2-6x}+x+2\right)}\)
\(=\dfrac{3\cdot2+2}{\left(2+3\right)\left(\sqrt{4\cdot2^2-6\cdot2}+2+2\right)}\)
\(=\dfrac{8}{5\cdot6}=\dfrac{8}{30}=\dfrac{4}{15}\)
