$$\lim_{x \to +\infty } (x+2) {\sqrt{\dfrac{x-1}{x^3 + x}}}$$ $$= \lim_{x \to +\infty } {\sqrt{(x+2)^2 \cdot \dfrac{x-1}{x^3 + x}}}$$ $$= \lim_{x \to +\infty } {\sqrt{\dfrac{x^3+3x^2-4}{x^3 + x}}}$$ $$= \lim_{x \to +\infty } {\sqrt{\cfrac{1 + \cfrac{3}{x} - \cfrac{4}{x^3}}{1 + \cfrac{1}{x^2}}}}$$ $$= \sqrt{\dfrac{1}{1}} = 1$$