Approaching Infinity

Pre-Calculus · Axiom Academy

Some curves chase a value forever and never arrive. Watch one rocket off the top of the page near a forbidden x — then drive that runaway behavior yourself. A curve that runs off the top of the page Plug a number into and you always get an answer — except at x = 0 , where dividing by zero is forbidden. The interesting part is what happens as you approach that forbidden value: the output doesn't politely stop, it explodes. The graph shoots up without bound, hugging an invisible wall it never touches. That wall is a vertical asymptote , and it's the signature of a rational function. Watch the sweep line walk in toward x = 0 along the real graph of . Out near x = 4 the curve is calm and low; as the line closes in on the forbidden value, f(x) climbs faster and faster and pins against the ceiling. It never reaches the dashed wall — it just keeps approaching. As x slides toward the forbidden value, f(x) grows without bound — that's what "approaching infinity" means. Push x toward the forbidden value yourself Here is . The denominator is zero at x = 2 , so that's the forbidden value — the vertical asymptote. Drag the point along the real curve toward x = 2 and watch the output blow up: approach from the left and it dives to , approach from the right and it rockets to . The closer you get, the bigger |f(x)| — but the wall is never reached. Distance to the wall → 0 forces |f(x)| → . The two halves race off in opposite directions — that's a vertical asymptote.

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