Estimating Limit Values from Graphs

A limit asks where a curve is headed, not where it lands. To read one off a graph, put your finger on the curve to the left of the target x-value and slide right toward it, noting the height you're approaching. Then do the same from the right, sliding left. If both sides approach the same height, that shared height is the limit. The graph often marks this approached height with an open circle — a hole in the curve — because the function might not actually be defined there, or might be defined somewhere else entirely.

That's why a solid dot can mislead you. The solid dot shows the actual output of the function at that x-value, which graphs sometimes draw as a separate point, floating above or below the open circle. A limit doesn't care about that solid dot at all. It only cares about the trend of the curve as you approach, from both directions. The function's value and its limit agree when the function is continuous there, but a graph with a jump, hole, or relocated point is precisely showing you a case where they split apart.

This graphical reading sets up everything after it. Once you can trace curves by eye, you'll compare the left-hand and right-hand readings directly: if they disagree, the two-sided limit doesn't exist. If the curve rockets upward or downward without settling on a finite height, you'll call that behavior unbounded rather than assign it a numerical limit. Both ideas build straight out of this same habit of tracing in from each side.