[tex]\displaystyle \sf \lim_{ x \to 0} \left[\frac{\sqrt{4+x}-2}{x}\right] \\\\\\ \text{ao fazer x = 0 dar\'a indetermina\c c\~ao. Ent\~ao vamos manipular a express\~ao }. \\\\ \text{Racionalizando o numerador} : \\\\\ \lim_{ x \to 0} \left[\left(\frac{\sqrt{4+x}-2}{x}\right)\cdot \left(\frac{\sqrt{4+x}+2}{\sqrt{4+x}+2} \right) \right] \\\\\\ \lim_{ x \to 0} \left(\frac{\sqrt{4+x}^2-2^2}{x\cdot (\sqrt{4+x}+2)}\right) \to \lim_{ x \to 0} \left(\frac{4+x-4}{x\cdot (\sqrt{4+x}+2)}\right)[/tex]

[tex]\displaystyle \sf \lim_{ x \to 0} \left(\frac{1}{\sqrt{4+x}+2}\right) =\frac{1}{\sqrt{4+0}+2} = \frac{1}{4} \\\\\\ Portanto : \\\\ \Large\boxed{\sf \ \lim_{ x \to 0} \left[\frac{\sqrt{4+x}-2}{x}\right] = \frac{1}{4} \ }\checkmark[/tex]