Interactive guide
A limit asks what f(x) is heading toward, not what's there.
Drag either dot along the curve, toward the dashed line from either side.
approaching from the left approaching from the right x = c
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- f(x), left
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- f(x), right
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Neither dragged point is ever allowed to land exactly on c, that's the entire idea of a limit: it describes what f(x) is doing arbitrarily close to c, on either side, without ever asking what happens (or doesn't happen) exactly at c itself.
Three ways the two sides can disagree
A two-sided limit only exists when both one-sided approaches are heading to the same place. When they don't, there are really only three ways it can go wrong, and each one has a preset above.
When they DO agree, there's a second question left: does the function actually equal that value at c, or does it just approach it? A function can have a perfectly good limit at a point where it isn't even defined, that's exactly what a "hole" in a graph is.
Reading the four presets
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1
Jump discontinuity
Left and right limits are both finite numbers, but different ones. No two-sided limit exists.
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2
Removable discontinuity (a hole)
Both sides agree on a value, but f(c) itself is undefined. The limit exists; the function just isn't there to meet it.
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3
Infinite discontinuity
Both sides run off to infinity, in the same direction or opposite ones. Neither side settles on a number, so there's nothing for them to agree on.
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4
Continuous
Both sides agree, and f(c) equals what they agree on. No gap, no jump, no hole, the ordinary case.
This is exactly the gap a derivative has to survive: the slope of a secant line is a fraction that turns into 0/0 the instant the two points it's built from collide. The Derivatives & Integration page is that same one-sided-approach idea, aimed at a ratio instead of a function value, watch the secant line there for the exact picture this page draws in miniature, twice, once per side.
The jargons
What "approaching" means, and the handful of ways it can fail to land anywhere.
What f(x) approaches as x gets arbitrarily close to c from just one direction, written x→c⁻ (left) or x→c⁺ (right).
Exists only when both one-sided limits exist and agree. Written limx→c f(x).
Both one-sided limits are finite but unequal, the graph visibly steps from one height to another.
The limit exists, but f(c) is undefined (or doesn't match it), a single missing or misplaced point, fixable by just defining it there.
f(x) grows without bound approaching c, a vertical asymptote. Neither side has a finite limit.
The limit exists AND equals f(c). All three conditions have to hold; a function can fail continuity by missing any one of them.