$f(x)=\frac{x-\sqrt{4{{x}^{2}}+9x}}{3x+1}\Rightarrow \underset{x\to \pm \infty }{\mathop{\lim }}\,f(x)=\underset{x\to \pm \infty }{\mathop{\lim }}\,\frac{x-\sqrt{4{{x}^{2}}}\left( \sqrt{1+\frac{9}{4x}} \right)}{3x+1}$ $\Rightarrow \left\{ \begin{matrix}    \underset{x\to +\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,\frac{x-2x}{3x+1}=-\frac{1}{3}  \\    \underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to -\infty }{\mathop{\lim }}\,\frac{x+2x}{3x+1}=1  \\ \end{matrix}\Rightarrow \frac{1}{3}+1=\frac{2}{3} \right.$