$\displaystyle\lim_{x \to 0} \dfrac{\sqrt{x+1}-\sqrt{x^2+x+1}}{x}\\ =\displaystyle\lim_{x \to 0} \dfrac{(\sqrt{x+1}-\sqrt{x^2+x+1})(\sqrt{x+1}+\sqrt{x^2+x+1})}{x(\sqrt{x+1}+\sqrt{x^2+x+1})}\\ =\displaystyle\lim_{x \to 0} \dfrac{x+1-x^2-x-1}{x(\sqrt{x+1}+\sqrt{x^2+x+1})}\\ =\displaystyle\lim_{x \to 0} \dfrac{-x^2}{x(\sqrt{x+1}+\sqrt{x^2+x+1})}\\ =\displaystyle\lim_{x \to 0} \dfrac{-x}{\sqrt{x+1}+\sqrt{x^2+x+1}}\\ =\displaystyle\lim_{x \to 0} \dfrac{-0}{\sqrt{0+1}+\sqrt{0^2+0+1}}\\ =0$
