$\sqrt{x+2}-2 \sqrt{x+1} + \sqrt{x} = \sqrt{x+2} + \sqrt{x} -2 \sqrt{x+1}$
$=\frac{(\sqrt{x+2} + \sqrt{x} -2 \sqrt{x+1})(\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1})}{\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1}}= \frac{x+2+x+2 \sqrt{x+2} \sqrt{x}-4x-4}{\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1}} $
$= 2\frac{\sqrt{x} \sqrt{x+2} -(x+1)}{\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1}}=2\frac{(\sqrt{x} \sqrt{x+2} -(x+1))(\sqrt{x} \sqrt{x+2} +(x+1))}{(\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+2})(\sqrt{x} \sqrt{x+2} +(x+1))}=2 \frac{x(x+2)-(x+1)^2}{(\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+2})(\sqrt{x} \sqrt{x+2} +(x+1))} =2\frac{-1}{(\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1})(\sqrt{x} \sqrt{x+2} +(x+1))}$
$ \Rightarrow \lim_{x\to \infty } x^{ \frac{3}{2} }(\sqrt{x+2}-2 \sqrt{x+1} + \sqrt{x})$
$= \lim_{a\to b} 2\frac{-x^{ \frac{3}{2} }}{(\sqrt{x+2} + \sqrt{x} +2 \sqrt{x+1})(\sqrt{x} \sqrt{x+2} +(x+1))}$
$=\lim_{x\to \infty } 2\frac{-x^{ \frac{3}{2} }}{x\sqrt{x}( \sqrt{1+ \frac{2}{x} } +1 +2 \sqrt{1+ \frac{1}{x} })(\sqrt{1+ \frac{2}{x} } +(1+ \frac{1}{x} ))}=$
$\lim_{x\to \infty } 2\frac{-x^{ \frac{3}{2} }}{x\sqrt{x}( \sqrt{1+ \frac{2}{x} } +1 +2 \sqrt{1+ \frac{1}{x} })(\sqrt{1+ \frac{2}{x} } +(1+ \frac{1}{x} ))}$
$=\lim_{x\to \infty } 2\frac{-1}{( \sqrt{1+ \frac{2}{x} } +1 +2 \sqrt{1+ \frac{1}{x} })(\sqrt{1+ \frac{2}{x} } +(1+ \frac{1}{x} ))}= \frac{-2}{(1+1+2)(1+1)}= \frac{-1}{4} $
