\displaystyle \lim_{x\rightarrow 0}\frac{\sqrt{x+9}-x-3}{\sqrt{x+9}-3}=\lim_{x\rightarrow 0}\frac{(\sqrt{x+9}-x-3)(\sqrt{x+9}+3)}{(\sqrt{x+9}-3)(\sqrt{x+9}+3)}=\lim_{x\rightarrow 0}\frac{x-x(\sqrt{x+9}+3)}{x}