Calculus · §2.2
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Where a function is headed near a point — even when we can't, or shouldn't, plug it in.
The opening problem
f(x) = (x² − 4) / (x − 2)
Undefined at x = 2 — a hole. Where is f headed as x → 2?
The closing-in method
Both sides settle on 4 — without ever using x = 2.
Definition 2.2.1
lim (x→a) f(x) = L
The limit reports where the neighborhood is headed — not the value at a.
Estimate from a table
lim (x→0) sin x / x = 1
±0.1 → 0.998, ±0.01 → 0.99998 … both sides → 1.
Estimate from a graph
g(−1) = 4, but lim = 3
The limit follows the curve — not the isolated marked point.
When a limit fails
Oscillation never settles (DNE); blow-up → +∞. Each earns its own vocabulary.
One-sided limits · Theorem 2.2.2
lim (x→a−) f = lim (x→a+) f = L
Two-sided limit exists ⇔ the sides agree. Disagree → DNE.
Infinite limits
1/x → −∞ then +∞. Infinity names the behavior — a vertical asymptote.
§2.2 — conclusions
The limit always reads the neighborhood — never the point.