Horizontal Asymptotes

Pre-Calculus · Axiom Academy

The y-value a curve flattens toward as — decided entirely by comparing the degrees of the top and bottom. 1. The Curve Flattens Onto a Line Take . Near the origin it bends, but follow it outward: the graph presses closer and closer to the line y = 2 and never pulls away again. That limiting line is the horizontal asymptote — the value f(x) approaches as . The curve never reaches y=2 , but the gap shrinks to zero so y = 2 is the horizontal asymptote 2. Why Only the Leading Terms Survive Why does head to 2 ? Divide top and bottom by the highest power, x^2 . Every lower-degree piece becomes , which collapses to 0 as x grows. What is left is the ratio of the leading coefficients — here . You are left with — the asymptote. Lower degree on top numerator dies first . Higher on top it outgrows the bottom no limit. The leading-coefficient ratio only matters when the degrees match . If the bottom wins the degree race the whole fraction is pulled to 0 ; if the top wins, it runs off without bound. The degree comparison decides which of those three things happens. 3. Three Functions, Three Fates Plot one example of each case on the same axes and sweep them all to the right. They split apart: one presses onto y=0 , one presses onto y=3 , and one escapes off the top of the frame entirely — no horizontal asymptote at all . Bottom wins (green): has degree 1 over degree 2 . The denominator grows faster, dragging the fraction down to y = 0 .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.