Limits of Rational Functions at Infinity (Slider-Controlled Numerator Degree)

① This example uses a slider p to change the degree of the numerator of a rational function, visually demonstrating the three typical outcomes as x approaches positive or negative infinity: a finite limit, divergence to ±∞, or no limit. ② The core mathematical concept is the limit of rational functions at infinity, focusing on how the comparison between the highest-degree terms of the numerator and denominator (equal degrees, lower numerator degree, higher numerator degree) determines the limit and the existence of horizontal asymptotes. ③ Construction highlights: the GeoGebra Limit command is used to compute limits, with objects p (slider), f (rational function), a (limit value), g (auxiliary function), h (horizontal asymptote), and text1/text2 (text displays of limit results), all updated interactively via the slider. ④ Suitable for senior high school or early university calculus students, this example aids in understanding the intuitive meaning of limits at infinity and serves as a classroom demonstration tool to connect algebraic degree comparisons with geometric asymptotes.