Limits at Infinity

Calculus Readiness · Axiom Academy

What value does a function settle toward as x runs off to the right or left forever? That settling value is its horizontal asymptote. 1. The Curve Flattens Onto a Line Watch a point ride the curve as x marches far to the right . The curve doesn't shoot off and it doesn't wander — it presses down toward the flat dashed line y = 2 and hugs it. That line is the horizontal asymptote , and the height it settles at is the limit at infinity. As x runs off to the right, the height settles… …onto the horizontal asymptote y = 2 means f(x) gets arbitrarily close to L once x is large enough. Replacing with asks the same question off to the left . The line y = L is then a horizontal asymptote — and a function can have up to two, one for each direction. 2. The Degree Decides the Asymptote For with P, Q polynomials, the far-out behavior depends only on the highest-degree term on top and bottom. Comparing those two degrees sorts every rational function into exactly three cases — watch three curves below each find their fate as . The bottom wins, so the fraction is crushed to nothing: . Horizontal asymptote y = 0 . Neither wins; the limit is the ratio of leading coefficients . Horizontal asymptote at that number. The top wins and runs away: . No horizontal asymptote. Far out, 3x^2 + 5x - 1 behaves just like 3x^2 — the smaller-degree terms become a rounding error.

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