Do \(x\rightarrow-\infty\Rightarrow\left|x\right|=-x\)
a. \(lim_{x\rightarrow-\infty}\dfrac{\left|2x\right|^3-\left|x\right|+1}{4\left|x^3\right|+x^2+1}=lim_{x\rightarrow-\infty}\dfrac{-8x^3+x+1}{-4x^3+x^2+1}\)\(=lim_{x\rightarrow-\infty}\dfrac{-8+\dfrac{1}{x^2}+\dfrac{1}{x^3}}{-4+\dfrac{1}{x}+\dfrac{1}{x^3}}=2\)
b. \(lim_{x\rightarrow-\infty}\dfrac{\left|x\sqrt{x^2+3}+1\right|}{\left|x^2-1\right|+x}=lim_{x\rightarrow-\infty}\dfrac{\left|\dfrac{\sqrt{x^2+3}}{x}+\dfrac{1}{x^2}\right|}{\left|1-\dfrac{1}{x^2}\right|+\dfrac{1}{x}}\) \(=lim_{x\rightarrow-\infty}\dfrac{\left|\dfrac{1}{x^2}-\sqrt{1+\dfrac{3}{x^2}}\right|}{\left|1-\dfrac{1}{x^2}\right|+\dfrac{1}{x}}=\dfrac{-1}{1}=-1\)
