In calculus, understanding limits at infinity is crucial, especially when dealing with rational functions. When tasked with finding the limit of a rational function as \( x \) approaches infinity, a systematic approach can simplify the process significantly. The first step involves identifying the highest power of \( x \) in both the numerator and the denominator.

To solve a limit algebraically, divide every term in the rational function by the highest power of \( x \) present in the denominator. For example, consider a rational function where the highest power in the denominator is \( x^3 \). By dividing each term in both the numerator and denominator by \( x^3 \), the limit can be rewritten in a simpler form. As \( x \) approaches infinity, any term with \( x \) in the denominator will approach zero. Thus, if the highest power in the denominator exceeds that in the numerator, the limit will converge to zero:

\[\lim_{{x \to \infty}} \frac{3x^2}{x^3} = \frac{3}{x} \to 0\]

Conversely, if the highest power in the numerator is greater than that in the denominator, the limit will diverge to either positive or negative infinity, indicating that the limit does not exist. For instance, if the numerator has a degree of 4 and the denominator has a degree of 2, the limit will approach infinity:

\[\lim_{{x \to \infty}} \frac{x^4}{x^2} \to \infty\]

In cases where the degrees of the numerator and denominator are equal, the limit can be determined by taking the ratio of the leading coefficients. For example, if the leading coefficient of the numerator is 2 and that of the denominator is 5, the limit evaluates to:

\[\lim_{{x \to \infty}} \frac{2x^1}{5x^1} = \frac{2}{5}\]

This method is effective for both positive and negative infinity, providing a reliable shortcut for evaluating limits of rational functions. Remember, if the degree of the denominator is higher, the limit is zero; if the degree of the numerator is higher, the limit does not exist; and if the degrees are equal, the limit is the ratio of the leading coefficients. Mastering these concepts will enhance your ability to tackle limits at infinity with confidence.