$$\eqalign{ \mathop {\lim }\limits_{x \to 0} \frac{{\ln \frac{{\sqrt {x + 1} + 1}}{2}}}{{{\mathop{\rm arctg}\nolimits} 4x}} } $$

\[\mathop {\lim }\limits_{x \to 0} \frac{{{a^x} + {b^x} - 2}}{{{c^x} + {d^x} - 2}}\]

\[\mathop {\lim }\limits_{x \to 0} \frac{{\cos \left( {4x} \right) - \cos x}}{{\cos \left( {2x} \right) - \cos x}}\]

\[\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \left( {\operatorname{tg} 3x\operatorname{tg} 8x} \right)\]