$\mathop {\lim }\limits_{x \to 0} f(x) = \mathop {\lim }\limits_{x \to 0} \frac{{ - x}}{{\sin x}} = \mathop {\lim }\limits_{x \to 0} \frac{{ - 1}}{{\frac{{\sin x}}{x}}} = \frac{{\mathop {\lim }\limits_{x \to 0} - 1}}{{\mathop {\lim }\limits_{x \to 0} \frac{{\sin x}}{x}}} = \frac{{ - 1}}{1} = - 1$