\(\displaystyle\lim_{x\to+\infty}\:\frac{x+\sqrt{x^2+1}}{x}-\lim_{x\to-\infty}\:\frac{x+\sqrt{x^2+1}}{x}\\\displaystyle\lim_{x\to+\infty}\:\frac{x+\sqrt{x^2(1+\frac{1}{x^2})}}{x}-\lim_{x\to-\infty}\:\frac{x+\sqrt{x^2(1+\frac{1}{x^2}})}{x}\\\displaystyle\lim_{x\to\infty}\:\frac{x+x\sqrt{1+\frac{1}{x^2}}}{x}-\lim_{x\to-\infty}\:\frac{x-x\sqrt{\frac{1}{x^2}}}{x}\\\displaystyle\lim_{x\to+\infty}\:\left(1+\sqrt{1+\frac{1}{x^2}}\right)-\lim_{x\to+\infty}\:\left(1-\sqrt{1+\frac{1}{x^2}}\right)=1\sqrt1-(1-\sqrt1)=2\)