<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] $$\begin{align}
\lim_{x\to 0}\frac{2\sqrt{x+1}-x-2}{x^2}
&= \lim_{x\to 0}\frac{2\sqrt{x+1}-(x+2)}{x^2} \frac{2\sqrt{x+1}+(x+2)}{2\sqrt{x+1}+x+2}\\
&= \lim_{x\to 0}\frac{4(x+1)-(x+2)^2}{x^2}  \frac{1}{2\sqrt{x+1}+x+2}\\
&= \lim_{x\to 0}\frac{4(x+1)-(x^2+4x+4)}{x^2}  \frac{1}{2\sqrt{x+1}+x+2}\\
&= \lim_{x\to 0}\frac{-x^2}{x^2}\frac{1}{2\sqrt{x+1}+x+2}\\
&=\lim_{x\to 0}\frac{-1}{2\sqrt{x+1}+x+2}\\
&=-\frac{1}{4}
\end{align}$$
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