
   $$
      \lim_{x\to+\infty}\frac{\sqrt{x+2} - \sqrt{x+1}}
                             {\sqrt{x+3} - \sqrt{x+1}} =
      \lim_{x\to0_+}\frac{\sqrt\frac{2x+1}x - \sqrt\frac{x+1}x}
                         {\sqrt\frac{3x+1}x - \sqrt\frac{x+1}x} =
      \lim_{x\to0_+}\frac{\sqrt{2x+1} - \sqrt{x+1}}
                         {\sqrt{3x+1} - \sqrt{x+1}}
   $$
   $$
    = \lim_{x\to0_+}\left(\frac{\sqrt{2x+1} - \sqrt{x+1}}
                               {\sqrt{3x+1} - \sqrt{x+1}}
                    \cdot \frac{\sqrt{2x+1} + \sqrt{x+1}}
                               {\sqrt{2x+1} + \sqrt{x+1}}
                    \cdot \frac{\sqrt{3x+1} + \sqrt{x+1}}
                               {\sqrt{3x+1} + \sqrt{x+1}}\right)=
   $$
   $$
      = \lim_{x\to0_+}\frac{x(\sqrt{3x+1} + \sqrt{x+1})}
                          {2x(\sqrt{2x+1} + \sqrt{x+1})} =
                      \frac{\sqrt1 + \sqrt1}{2(\sqrt1 + \sqrt1)} =
                      \frac12.
   $$