$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,\frac{\left[ x \right]}{x}=\frac{\left[ {{0}^{-}} \right]}{{{0}^{-}}}=\frac{-1}{{{0}^{-}}}=\frac{1}{{{0}^{+}}}=+\infty $