Gold beads move along a line toward an empty ring. A chip says heights; the last two beads are marked 9.9 and 9.99, and the ring is marked limit 10. Off the path, straight below the ring, sits an orange ball marked value, f(5) = 4.

Where are you heading? To the jar. That is the limit, even if someone moved the jar, and even if the spot is empty when you arrive. The limit is about the path, not about what sits at the end.

A pink curve on axes x and y dips to an open circle at x tick 1, height 3, marked limit. A filled gold dot above it, at height 5, is marked value.

The path heads to the open circle. The filled dot is the value at that x. They need not match. (On this teaching screen the path heads to height 3, and the value is 5.)