Infinite Limits & Vertical Asymptotes

Calculus Readiness · Axiom Academy

When a function blows up near a point: divide by something tiny and the output runs off to , hugging a vertical asymptote. Look at near x=0 . The denominator x^2 is a tiny positive number on either side of 0 , so dividing 1 by it gives a huge positive number. As the curve runs off the top of the page — and it does so on both sides. The output grows without bound as The line x=0 is a vertical asymptote 2. When the Two Sides Disagree Now . The denominator is no longer always positive: just right of 0 , x is a tiny positive number, so 1/x is huge and positive ; just left of 0 , x is a tiny negative number, so 1/x is huge and negative . The sign flips with the side. x is a tiny positive number, so . The branch climbs the asymptote upward. x is a tiny negative number, so . The branch plunges down the asymptote. The sides run in opposite directions, so does not exist — not even as . One infinite one-sided limit is enough: x=0 is a vertical asymptote of 1/x all the same. Don't memorize the direction — test it. To find , plug in a value just above 2 (like 2.01 , giving 1/0.01 = +100 ) and just below (like 1.99 , giving 1/(-0.01) = -100 ). The sign of the test value tells you whether you head to or . 3. Not Every Zero Denominator Blows Up A zero in the denominator is only a candidate for a vertical asymptote. Take . The denominator is zero at x=3 — but the numerator is zero there too ( 3^2-9=0 ). Factor and the trouble cancels :

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