1: \(\lim_{x\to-\infty}\frac{\left|x\right|+\sqrt{x^2+10}}{2x+3}\)
\(=\lim_{x\to-\infty}\frac{-x+\sqrt{x^2+10}}{2x+3}=\lim_{x\to-\infty}\frac{-x+\left(-x\right)\cdot\sqrt{1+\frac{10}{x^2}}}{2x+3}\)
\(=\lim_{x\to-\infty}\frac{-1-\sqrt{1+\frac{10}{x^2}}}{2+\frac{3}{x}}=\frac{-1-1}{2}=-\frac22=-1\)
2: \(\lim_{x\to-\infty}\left(\sqrt[3]{x^3+x}-x\right)\)
\(=\lim_{x\to-\infty}\frac{x^3+x-x^3}{\sqrt[3]{\left(x^3+x\right)^2}+x\cdot\sqrt[3]{x^3+x}+x^2}\)
\(=\lim_{x\to-\infty}\frac{x}{\sqrt[3]{\left(x^3+x\right)^2}+x\cdot x\cdot\sqrt[3]{1+\frac{1}{x^2}}+x^2}\)
\(=\lim_{x\to-\infty}\frac{x}{x^2\cdot\sqrt[3]{\left(1+\frac{1}{x^2}\right)^2}+x^2\cdot\sqrt[3]{1+\frac{1}{x^2}}+x^2}=\lim_{x\to-\infty}\frac{1}{x\cdot\sqrt[3]{\left(1+\frac{1}{x^2}\right)^2}+x\cdot\sqrt[3]{1+\frac{1}{x^2}}+x}\)
=-∞
3: \(\lim_{x\to-\infty}\left(\sqrt{4x^2+x+3}+2x+1\right)\)
\(=\lim_{x\to-\infty}\frac{4x^2+x+3-\left(2x+1\right)^2}{\sqrt{4x^2+x+3}-2x-1}=\lim_{x\to-\infty}\frac{-3x+2}{\sqrt{4x^2+x+3}-2x-1}\)
\(=\lim_{x\to-\infty}\frac{-3x+2}{-x\cdot\sqrt{4+\frac{1}{x}+\frac{3}{x^2}}-2x-1}=\lim_{x\to-\infty}\frac{-3+\frac{2}{x}}{-\sqrt{4+\frac{1}{x}+\frac{3}{x^2}}-2-\frac{1}{x}}\)
\(=\frac{-3+0}{-2-2}=\frac{-3}{-4}=\frac34\)
4: \(\lim_{x\to+\infty}\frac{x^4-x^3+11}{2x-7}=\lim_{x\to+\infty}\frac{x^3-x^2+\frac{11}{x}}{2-\frac{7}{x}}=+\infty\)
