\[ \begin{split} &\lim_{x \to 0} \frac{\sqrt{x+4}-2}{\sin 5x}=\left[\frac{0}{0}\right]=\\[6pt] &=\lim_{x \to 0} \frac{(\sqrt{x+4}-2)5x}{5x\cdot \sin 5x}=\\[6pt] &=\lim_{x \to 0}\frac{\sqrt{x+4}-2}{5x}=\left[\frac{0}{0}\right]^H=\\[6pt] &=\lim_{x \to 0}\frac{\dfrac{1}{2\sqrt{x+4}}}{5}=\\[6pt] &=\lim_{x \to 0}\frac{1}{10\sqrt{x+4}}=\\[6pt] &=\frac{1}{20} \end{split} \]