One-sided limits

You can approach a point from two directions. The left-hand limit watches f(x) as x rises toward a from below; the right-hand limit watches it fall toward a from above. Write them f(a⁻) and f(a⁺).

The two-sided limit exists only when both sides agree. If the graph jumps — the left side heads for 3 but the right side heads for 5 — there is no single limit at that point. Mismatched sides are how calculus detects a jump.