Limits of Polynomial Fractions

Rule

For limits in the form ${\lim }_{x\to \pm \infty }\frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials in $x$. \lim_{x \to \pm \infty} \frac{P(x)}{Q(x)}$$$$= \lim_{x \to \pm \infty}\frac{Ax^\alpha + ...}{Bx^\beta + ...} = \begin{cases} 0 & \text{if }\alpha < \beta \\ \frac{A}{B} & \text{if }\alpha = \beta\\ \infty \ \text{or} \ -\infty & \text{if }\alpha > \beta \end{cases} If $\alpha >\beta$, the sign of the limit depends on the sign of $A$ and $B$

Example

${\lim }_{x\to \infty }\frac{(18x^2+5x-1)(2\sqrt{x}-1{)}^3}{(3x-1{)}^4}$ $={\lim }_{x\to \infty }\frac{144x^{\frac{7}{2}}+...}{81x^4+...}=0$