Reading limits from graphs

To read $\lim\limits_{x \to a} f(x)$ from a graph, we ignore whatever is drawn at the point $x = a$ and trace the curve inward from both sides. If both sides close in on the same height $L$, that height is the limit. A graph marks what the function does at $a$ itself with a small open or closed dot.

Any dot at $x = a$, open or closed, records only the value $f(a)$. It has no bearing on the limit, which is defined entirely by the curve's behaviour from each side. This lets us read three situations cleanly, with the limit equal to $L$ in each:

1. An open dot at $(a, L)$ leaves $f(a)$ undefined, while the limit is still $L$.
2. A closed dot at some other height fixes $f(a)$ at that height, while the limit is still $L$.
3. A closed dot lying on the curve gives $f(a) = L$, so $\lim\limits_{x \to a} f(x) = f(a)$.

Use the interactive diagram below to see an example of each case.

A limit exists only when the curve closes in on the same finite height from both sides. When that fails, the limit does not exist. On a graph, three patterns cause this: a jump, where the two sides settle on different heights; an unbounded curve, which runs off without limit; and oscillation, where the curve wobbles faster and faster without ever settling.

As we've seen, a hole is not one of these patterns. If the curve has a visible gap at $x = a$, but both sides still close in on the same height, the limit exists. A limit fails only when the curve's trend breaks, not when a single point is missing.