<s>[INST] Find $\displaystyle\lim_{x\to\infty}x - \sqrt{x+1}\sqrt{x+2}$ using squeeze theorem 
Find $$\lim_{x\to\infty}x - \sqrt{x+1}\sqrt{x+2}$$ using squeeze theorem

Tried using binomial expansion, but have no idea on how to continue.
 [/INST] $$\begin{align*}&x-\sqrt{x+1}\sqrt{x+2}=\frac{x^2-(x+1)(x+2)}{x+\sqrt{x+1}\sqrt{x+2}}=\frac{-3x+2}{x+\sqrt{x^2+3x+2}}=\\{}\\
&\frac{-3+\frac2x}{1+\sqrt{1+\frac3x+\frac2{x^2}}}\xrightarrow[x\to\infty]{}\frac{-3+0}{1+\sqrt{1+0+0}}=-\frac32\end{align*}$$
</s>