Vertical Asymptotes

Pre-Calculus · Axiom Academy

The invisible walls a rational function races toward but can never cross. 1. A Curve Racing Toward a Wall Take the simplest interesting rational function, . At x = 2 the denominator is 0 , so the function is undefined there. Watch what happens as x creeps toward 2 : the output explodes. Approaching from the right the curve shoots up to ; from the left it plunges to . The dashed red line x = 2 is the vertical asymptote — a wall the curve hugs ever tighter but never touches. Denominator zero at x = 2 → vertical asymptote You don't have to graph a function to find its vertical asymptotes — you read them off the denominator. Here's the procedure, applied to . Each wall splits the plane, and the curve can dive a different way on each side. Near x = -2 the left branch climbs to while the middle dives to ; near x = 2 the middle dives to and the right branch climbs to . The denominator's sign change is what flips the curve from one infinity to the other. Not every denominator zero is a vertical asymptote. Consider . The denominator vanishes at x = 3 and x = -1 — but x = 3 also makes the numerator zero. That shared factor (x-3) cancels, leaving . So x = 3 is just a hole (the curve sails through at height , with one point missing), and the only true vertical asymptote is x = -1 . Numerator and denominator both hit zero. The factor cancels; the curve is finite here (value ) with a single point punched out. No infinity.

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