<s>[INST] Algebraically Solve Limit $$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}$$
I can solve it using l'Hôpital but just cannot find a way to do it algebraically.
 [/INST] We have $\sqrt{1+x}=1+\frac{1}{2}x+(\frac{1}{2})(\frac{-1}{2})\frac{x^2}{2}+o(x^2)$, so
$$\lim_{x \to 0} \dfrac{2\sqrt{x+1}-x-2}{x^2}=\lim_{x \to 0}\dfrac{-\frac{1}{4}x^2+o(x^2)}{x^2}=-\frac{1}{4}$$
</s>