$\underset{x\to 0}{Lt}\frac{\sqrt{3+x}-\sqrt{3}}{x}=$

$\underset{x\to 0}{Lt}\frac{\sqrt{3+x}-\sqrt{3}}{x}$

$=\underset{x\to 0}{Lt}\frac{\sqrt{3+x}-\sqrt{3}}{x}\times \frac{(\sqrt{3+x}+\sqrt{3})}{(\sqrt{3+x}+\sqrt{3})}$

$=\underset{x\to 0}{Lt}\frac{{\left(\sqrt{3+x}\right)}^2-{\left(\sqrt{3}\right)}^2}{\left(x\right)(\sqrt{3+x}+\sqrt{3})}=\underset{x\to 0}{Lt}\left(\frac{3+x-3}{\left(x\right)(\sqrt{3+x}+\sqrt{3})}\right)$

$=\underset{x\to 0}{Lt}\frac{x}{x(\sqrt{3+x}+\sqrt{3})}=\frac{1}{\sqrt{3}+\sqrt{3}}$

$=\frac{1}{2\sqrt{3}}$.