The Idea of a Limit

This limit calculator evaluates the behavior of a function as its input approaches a target point from both sides. The left-hand limit samples values just below the point, the right-hand limit just above, and the two-sided limit exists only when they agree.

Continuity and Discontinuities

When the limit equals f(a), the function is continuous there. A jump (one-sided limits differ), a hole (removable discontinuity), or an infinite blow-up all produce characteristic limit patterns the calculator identifies and labels.

Special Indeterminate Forms

At 0/0, direct substitution fails. The curated library resolves the classic cases exactly — sin x / x → 1 and (eˣ − 1) / x → 1 at zero — while the curve plot makes the approach visible.