Welcome to Limits

Calculus Readiness · Axiom Academy

The one idea that makes calculus possible: as x creeps toward a number, where is the function headed? You've reached the doorway to calculus You've rebuilt algebra, mastered functions, and worked through trigonometry — all of it "static" math that asks what a quantity is . Calculus asks something new: how things change , and what they're heading toward . Every bit of it — every derivative, every integral — is built on one idea you can actually watch happen: the limit . A limit asks a simple question: as x creeps closer and closer to some number a , what value does f(x) approach? Watch a point slide in along the curve from the left and another from the right . Both heights close in on the same value L . That shared destination is the limit: . Notice nobody ever plugs in x = a . We just watch where the function is going — and from both sides it's going to the same place. Slide the dial to shrink the distance to a . Two points walk in along the curve — one from each side — and their heights squeeze together toward L . Watch the gap melt toward 0 : that's f(x) getting "closer and closer." Getting close is the whole game; you never have to land exactly on a . As the distance to a goes to 0 , f(x) closes on L from both sides. Same destination, both ways in — that's what means. The limit isn't the same as the value at a

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