Curve Footer

Limit Trace Curve is a calculus visualisation game that teaches the concept of limits the way no textbook can — by showing a particle approach a point on a function curve in real time, and asking you to predict where it is heading before it gets there. The limit of a function is not the value at a point. It is the value the function approaches as the input gets close to that point. The distinction matters because the most important cases in calculus are precisely the ones where the function is not defined at the point, or behaves unexpectedly there. A particle that slows as it approaches a target, halving the remaining distance each frame, makes this distinction physical rather than abstract. The function curve is rendered in a gradient that tracks the function's slope — violet where the curve falls steeply, teal where it flattens, amber where it rises. The particle follows the curve and leaves a luminous trail. An approach table updates in real time, showing f(x) for values closer and closer to the target. When the particle halts at its minimum distance, the canvas zooms in automatically — a hundredfold — to reveal what is actually happening near the limit point. A smooth curve continues without interruption. An open circle marks a removable discontinuity. Two curve segments separated by a gap mark a jump. A curve shooting vertically marks an asymptote. Ten levels cover the complete limit curriculum. The removable discontinuity level shows (x²−4)/(x−2) at x=2 — a curve that looks continuous at normal scale but reveals a tiny open circle at 100× zoom, making the difference between a limit and a function value visible for the first time. The squeeze theorem level shows three simultaneous curves closing like a vice, forcing an oscillating function to converge. The epsilon-delta level gives the player two sliders controlling ε and δ, and asks them to keep the curve inside both bands simultaneously — the formal definition made interactive rather than symbolic. The L'Hôpital level shows the original function failing to converge alongside the derivative ratio converging cleanly, in three panels. The particle's pitch tracks f(x) as it moves. The approach table plays a converging melody. Every visual has a corresponding sound.