Answer
$\lim_{x\to-\infty}\dfrac{x^{2}+2}{x^{3}+x+1}=0$
Work Step by Step
$\lim_{x\to-\infty}\dfrac{x^{2}+2}{x^{3}+x+1}$
Since $\lim_{x\to-\infty}f(x)=\lim_{x\to\infty}f(-x)$, substitute $x$ by $-x$ in the function and simplify:
$f(-x)=\dfrac{(-x)^{2}+2}{(-x)^{3}-x+1}=\dfrac{x^{2}+2}{-x^{3}-x+1}$
Evaluate $\lim_{x\to\infty}f(-x)$:
$\lim_{x\to\infty}\dfrac{x^{2}+2}{-x^{3}-x+1}$
Divide the numerator and the denominator by $x^{3}$:
$\lim_{x\to\infty}\dfrac{x^{2}+2}{-x^{3}-x+1}=\lim_{x\to\infty}\dfrac{\dfrac{x^{2}+2}{x^{3}}}{\dfrac{-x^{3}-x+1}{x^{3}}}=...$
Simplify:
$...=\lim_{x\to\infty}\dfrac{\dfrac{x^{2}}{x^{3}}+\dfrac{2}{x^{3}}}{\dfrac{-x^{3}}{x^{3}}-\dfrac{x}{x^{3}}+\dfrac{1}{x^{3}}}=\lim_{x\to\infty}\dfrac{\dfrac{1}{x}+\dfrac{2}{x^{3}}}{-1-\dfrac{1}{x^{2}}+\dfrac{1}{x^{3}}}=...$
Apply direct substitution:
$...=\dfrac{\dfrac{1}{\infty}+\dfrac{2}{\infty^{3}}}{-1-\dfrac{1}{\infty^{2}}+\dfrac{1}{\infty^{3}}}=\dfrac{0+0}{-1-0+0}=\dfrac{0}{-1}=0$
