AP® Calculus BC · Unit 1 · Topic 1.3
Estimating Limit Values from Graphs
Reading a limit from a graph means tracing the curve toward an x-value from both sides and seeing what height it approaches, ignoring where the function is actually plotted there.
A limit is about approach, not arrival. To read one off a graph, put your finger on the curve far to the left of the x-value in question and slide it rightward, watching the height your finger climbs toward. That's the left-hand limit. Then do the same sliding in from the right — that's the right-hand limit. If both slides land on the same height, that shared height is the limit, even if the graph has an open circle there showing the function is undefined or defined somewhere else at that exact point. The open circle tells you about the function's value; the limit only cares about the trend.
This is why a solid dot sitting apart from an open circle doesn't change the answer. The curve approaches the open circle's height from both sides, so that's the limit, full stop. The solid dot is just f(c), a separate fact about the function's actual output, and the two numbers are allowed to disagree.
A jump discontinuity is different: the left and right slides land on different heights, so there's no single number both sides agree on, and the two-sided limit does not exist there. Picking the taller side is a tempting shortcut but it isn't how limits work — disagreement means no limit, period. This graphical reading is the same judgment a table of values makes numerically: plugging in x-values that creep toward c from below and above and watching whether the outputs converge. Graphs make the approach visible; tables make it numerical. Both set up the formal definition of continuity coming next, which demands the limit exist and match f(c) exactly.
Key terms
- One-sided limit
- The height a curve approaches as x comes in from only the left or only the right of a point.
- Two-sided limit
- The height a curve approaches from both directions at once; it exists only if both one-sided limits agree.
- Open circle
- A graph marking showing a height the function approaches but does not actually equal at that x-value.
- Jump discontinuity
- A break in a graph where the left-hand and right-hand limits approach different heights.
Watch it explained
- How it worksHow do I trace a curve inward from the left and the right to read a one-sided limit?
- Worked exampleFind the limit as x approaches 1 for a graph with an open circle at (1, 3) and a solid dot at (1, 5).
- Common mistakeWhy isn't a jump in a graph a place where the limit equals the taller piece?
- Ties togetherHow does reading this limit off a graph connect to reading the same limit off a table of values?
Check yourself
End of topic.