[tex]\lim_{x \to \pm\infty}\limits{\Big(\dfrac{x^2}{x+1}-x\Big) } = \lim_{x \to \pm\infty}\limits{\Big(\dfrac{x^2}{x+1}-\dfrac{x(x+1)}{x+1}\Big) } = \\ \\ = \lim_{x \to \pm\infty}\limits{\Big(\dfrac{x^2-x(x+1)}{x+1}\Big) } = \lim_{x \to \pm\infty}\limits{\Big(\dfrac{x^2-x^2-x}{x+1}\Big) } = \\ \\ = \lim_{x \to \pm\infty}\limits{\dfrac{-x}{x+1}} = \lim_{x \to \pm\infty}\limits{\dfrac{-\not{x}}{\not{x}\Big(1+\dfrac{1}{x}\Big)} = \lim_{x \to \pm\infty}\limits{\dfrac{-1}{1+\dfrac{1}{x}}} =[/tex]
[tex]=\dfrac{-1}{1+\dfrac{1}{\pm\infty}} = \dfrac{-1}{1+0} = \dfrac{-1}{1} = -1[/tex]
