Mastering Limits at Infinity and Horizontal Asymptotes

Miacademy & MiaPrep Learning Channel

This educational video provides a comprehensive guide to understanding limits at infinity, a fundamental concept in Precalculus and Calculus. It bridges the gap between graphical intuition and algebraic rigor by connecting limits at infinity directly to the concept of horizontal asymptotes. The video progresses from visual inspection of rational function graphs to formal definitions using limit notation, and finally to analytic methods for evaluating limits without graphing calculators. The content explores key themes such as end behavior, rational functions, and the algebraic manipulation required to solving limits. It specifically addresses how the degrees of the numerator and denominator influence the limit, covering three main cases: when the denominator's degree is higher (limit is 0), when degrees are equal (limit is the ratio of leading coefficients), and when the numerator's degree is higher (limit is infinite or DNE). It also touches upon special cases involving oscillating trigonometric functions and exponential functions. For educators, this video is an excellent resource for introducing or reinforcing the concept of end behavior in a rigorous way. It moves beyond simple memorization of rules by explaining the "why" behind them through the technique of dividing by the highest power of x. The clear step-by-step examples make it suitable for flipped classrooms, review sessions, or as a primary instructional tool for introducing the analytic evaluation of limits.