$$\eqalign{ \mathop {\lim }\limits_{x \to \infty } \left( {\left( {{x^3} + x} \right) \cdot \sin \frac{1}{x} - {{\left( {{x^6} - 5{x^4} + 1} \right)}^{\frac{1}{3}}}} \right) } $$

$$\eqalign{ \mathop {\lim }\limits_{x \to 0} {\left( {\frac{{\sin x}}{{2\left( {\sqrt {x + 1} - 1} \right)}}} \right)^{{\mathop{\rm ctg}\nolimits} x}} } $$

\[\mathop {\lim }\limits_{x \to 0} \frac{{\operatorname{tg} \left( {\sin x} \right) - \sin \left( {\operatorname{tg} x} \right)}}{{{x^7}}}\]