Limits and Continuity: Complete Guide
Limits and continuity are the rigorous foundation of all calculus. Before derivatives and integrals can be defined, we need a precise language for describing what happens to a function as its input approaches a value.
A function f is continuous at x = a if the limit exists, equals the function value, and the function is defined there. Continuity is the bridge between the abstract limit concept and the "no holes or jumps" intuition.
Formula
lim[x→a] f(x) = L iff ∀ε>0, ∃δ>0: 0<|x−a|<δ → |f(x)−L|<ε
The Formal Definition of a Limit
Informal definition: lim[x→a]f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a (but not equal to a).
Formal (ε-δ) definition: for every ε > 0 (desired precision of output), there exists δ > 0 (sufficient precision of input) such that whenever 0 < |x−a| < δ, we have |f(x)−L| < ε. The limit is about the neighborhood around a, not the value at a.
Key point: the function need not be defined at x = a for the limit to exist. lim[x→0] sin(x)/x = 1, even though sin(0)/0 = 0/0 is undefined.
One-Sided Limits and When Limits Fail to Exist
Left-hand limit: lim[x→a⁻]f(x) = L means f(x)→L as x approaches a from below (x < a).
Right-hand limit: lim[x→a⁺]f(x) = L means f(x)→L as x approaches a from above (x > a).
The two-sided limit exists if and only if both one-sided limits exist AND are equal. If they differ, the limit does not exist (DNE).
A limit also fails to exist if f(x) oscillates (e.g., sin(1/x) near 0) or diverges to ±∞.
| Scenario | Left Limit | Right Limit | Two-Sided Limit |
|---|---|---|---|
| f(x) = x² | a² | a² | a² ✓ |
| f(x) = |x|/x | −1 | +1 | DNE |
| f(x) = 1/x at 0 | −∞ | +∞ | DNE |
| f(x) = ⌊x⌋ at 2 | 1 | 2 | DNE (jump) |
| f(x) = sin(1/x) at 0 | DNE | DNE | DNE (oscillation) |
Continuity: Definition and Types of Discontinuity
A function f is continuous at x = a if three conditions hold: (1) f(a) is defined, (2) lim[x→a]f(x) exists, (3) lim[x→a]f(x) = f(a). Fail any one → discontinuity at a.
Removable discontinuity: the limit exists but either f(a) is undefined or f(a) ≠ limit. "Hole" in the graph. Example: f(x) = (x²−1)/(x−1) at x=1 (limit = 2, but undefined).
Jump discontinuity: left and right limits both exist but are unequal. Step function, ⌊x⌋ at integers.
Infinite discontinuity: one or both one-sided limits are ±∞. Vertical asymptote. Example: f(x) = 1/x at x = 0.
Oscillation discontinuity: no one-sided limit exists because the function oscillates. Example: sin(1/x) near 0.
Continuous functions are "connected" — no jumps, holes, or asymptotes. Polynomials, exponentials, trig functions, and their compositions are continuous everywhere on their domains.
Limit Laws and the Squeeze Theorem
Limit laws allow algebraic manipulation: lim(f±g) = lim f ± lim g; lim(fg) = lim f · lim g; lim(f/g) = lim f / lim g (if lim g ≠ 0); lim(cˢf) = c · lim f; lim[f(x)ⁿ] = (lim f)ⁿ.
Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L, then lim f(x) = L. Used to prove lim[x→0] sin(x)/x = 1 by sandwiching between cos x and 1.
Limit composition: if lim[x→a]g(x) = L and f is continuous at L, then lim[x→a]f(g(x)) = f(L). This justifies evaluating limits by substitution when the outer function is continuous.
Intermediate Value Theorem
IVT: If f is continuous on [a,b] and N is any value strictly between f(a) and f(b), then there exists c ∈ (a,b) such that f(c) = N.
Intuition: a continuous function cannot skip values — it must pass through every intermediate value. If f(0) = −1 and f(1) = 3, then f must equal 0 somewhere in (0,1).
Application — existence of roots: if f(a) < 0 and f(b) > 0 (or vice versa) and f is continuous, there is at least one root in (a,b). This is the basis for bisection root-finding algorithms.