Rational functions are represented as the quotient of two polynomial functions.
The end behavior of these functions is determined by the degree of the polynomials involved.
For large input values, the polynomial with the higher degree will dominate the behavior of the rational function.
This means you should focus on the leading terms of the numerator and denominator.
If the numerator's polynomial has a higher degree, the rational function behaves like a nonconstant polynomial, possibly with a slant asymptote.
If the degrees are equal, the function approaches a constant value, indicating a horizontal asymptote.
If the denominator's polynomial has a higher degree, the function approaches zero, resulting in a horizontal asymptote at y = 0.
The cognitive trap here is ignoring the significance of the leading terms.
Misjudging which polynomial dominates can lead to incorrect predictions about the graph's behavior at extremes, causing errors in both reasoning and graphical representation.