_ x → 0 x + 2 x x ^ 2 + 2 x + 1 - ^ 2 x - x + 1 is

\lim_{x \to 0}\dfrac{x + 2\sin x}{\sqrt{{x}^{2} + 2\sin x + 1} - \sqrt{\sin x - x + 1}}$\left(\dfrac{0}{0}form\right)$

By using L'Hospital Rule,

$= \lim_{x \to 0}\dfrac{1 + 2\cos x}{\dfrac{1\left(2x + 2\cos x\right)}{2\sqrt{{x}^{2} + 2\sin x + 1}} - \dfrac{1\left(\sin 2x - 1\right)}{2\sqrt{\sin^{2}x - x + 1}}}$

$= \dfrac{1 + 2}{1 + \dfrac{1}{2}} = \dfrac{3}{\dfrac{3}{2}} = 2$.