Limit of a Rational Fraction of Polynomials as x Tends to Infinity
\[f(x)\]
is given as a rational fraction of two polynomials, \[P(x), \; Q(x)\]
so that \[f(x)= \frac{P(x)}{Q(x)}\]
then we can evaluate the limit as \[x \rightarrow \infty\]
as in the following example.\[\begin{equation} \begin{aligned} lim_{x \rightarrow \infty} \frac{x^2-2x+3}{4x^2-x-1} &= lim_{x \rightarrow \infty} \frac{1-2/x+3/x^2}{4-1/x-1/x^2} \\ &= lim_{x \rightarrow \infty} \frac{1-0+0}{4-0-0} \\ &- \frac{1}{4} \end{aligned} \end{equation}\]
