$B=\lim_{x\to \infty}x\bigg(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x\bigg)$
$⇒B=\lim_{x\to \infty}x\bigg(\dfrac{2x^2+2x+2x\sqrt{x^2+2x}-4x^2-4x}{\sqrt{x^2+2x}+2\sqrt{x^2+x}+x}\bigg)$
$⇒B=\lim_{x\to \infty}x\bigg(2x.\dfrac{\sqrt{x^2+2x}-x-1}{\sqrt{x^2+2x}+2\sqrt{x^2+x}+x}\bigg)$
$⇒B=\lim_{x\to \infty}x\Bigg[\dfrac{-2x}{\big(\sqrt{x^2+2x}+2\sqrt{x^2+x}+x\big)\big(\sqrt{x^2+2x}+x+1\big)}\Bigg]$
$⇒B=\lim_{x\to \infty}x\dfrac{-2x^2}{\bigg(\sqrt{x^2+2x}+2\sqrt{x^2+x}+x\bigg)\bigg(\sqrt{x^2+2x}+x+1\bigg)}$
$⇒B=\lim_{x\to \infty}\dfrac{-2}{\Bigg(\sqrt{1+\dfrac 2x}+2\sqrt{1+\dfrac 1x}+1\Bigg)\Bigg(\sqrt{1+\dfrac 2x}+1+\dfrac 1x\Bigg)}$
$⇒B=-\dfrac 14$
