\(\displaystyle\lim_{x\to-\infty}\left(\sqrt{x^2+x}+x\right)=\displaystyle\lim_{x\to-\infty}\frac{\left(\sqrt{x^2+x}+x\right)\left(\sqrt{x^2+x}-x\right)}{\left(\sqrt{x^2+x}-x\right)}\;\\=\displaystyle\lim_{x\to-\infty}\frac{x^2+x-x^2}{\sqrt{x^2+x}-x}=\displaystyle\lim_{x\to-\infty}\frac{x}{\sqrt{x^2+x}-x}=\displaystyle\lim_{x\to-\infty}\frac{x}{\sqrt{x^2\left(1+\frac{1}{x}\right)}-x}\;\\=\displaystyle\lim_{x\to-\infty}\frac{x}{-x\sqrt{1+\frac{1}{x}}-x}=\displaystyle\lim_{x\to-\infty}\frac{1}{-\sqrt{1+\frac{1}{x}}-1}=\frac{1}{-1-1}=-\frac12\)