Limit Explorer

Limits rigorously formalize the concept of infinite topological proximity. By approaching a singular coordinate strictly without ever touching it, limits uniquely identify and categorize removable holes, infinite asymptotes, and non-removable jump discontinuities, establishing the exact bedrock foundation for all continuous calculus.

A limit asks where a function is heading as x closes in on a point — not what happens at the point itself. That gap matters: sin(x)/x is undefined at x = 0 (it reads 0/0), yet from both sides the graph aims straight at 1. Drag x toward the target and watch the left and right values agree; when they do, the limit exists even if the curve has a hole.

This tool samples f at x = 10⁻ⁿ for n = 1, 2, …, 8. Feed it sin(π/x) near 0 and every row reads exactly 0, because sin(10ⁿπ) = 0 for whole-number exponents — the table happily reports a limit of 0. Yet between those sample points the function swings between −1 and +1 forever, so the true limit does not exist. Numerical convergence is a hint, never a proof.