Connecting Limits at Infinity and Horizontal Asymptotes AB & BC

A limit at infinity describes end behavior, and a finite one is a horizontal asymptote. For rational functions the answer follows from comparing degrees: the limit is 0 when the numerator's degree is smaller, the ratio of leading coefficients when the degrees match, and unbounded when the numerator's is larger. The underlying method is to divide numerator and denominator by the highest power in the denominator, after which every remaining term with $x$ underneath tends to zero.

Two errors dominate. The degree rules get applied from memory without checking which case is in play, most often dividing leading coefficients when the degrees are not equal, and a graph gets rejected for crossing its horizontal asymptote, which is perfectly legal since the asymptote only describes far-out behavior. Separately, the limit laws get pushed past their hypotheses: $\frac{\infty}{\infty}$ and $\infty - \infty$ are indeterminate and need real work, and the two ends must be computed separately whenever an even root or absolute value is present.