$\underset{x\to -\infty }{\mathop{\lim }}\,\frac{3x+1}{2\left| x \right|-\sqrt{-x-1}}=\underset{x\to -\infty }{\mathop{\lim }}\,\frac{3x+1}{-2x-\sqrt{-x-1}}$ $=\underset{x\to -\infty }{\mathop{\lim }}\,\frac{x(3+\frac{1}{x})}{x(-2+\sqrt{\frac{-1}{x}-\frac{1}{{{x}^{2}}})}}=\underset{x\to -\infty }{\mathop{\lim }}\,\frac{3+\frac{1}{x}}{-2+\sqrt{\frac{-1}{x}-\frac{1}{{{x}^{2}}}}}=\frac{3+0}{-2+0}=-\frac{3}{2}$