1 · The limit idea and its notation
The expression lim x→a f(x) = L says: as x gets arbitrarily close to a, f(x) gets arbitrarily close to L. Two things are pointedly irrelevant: what f(a) equals (it may not even exist), and what happens exactly at a. Limits describe the journey, not the destination.
The rigorous version uses ε and δ: for every ε > 0 you can name a δ > 0 so that 0 < |x − a| < δ forces |f(x) − L| < ε. You rarely need to produce δ by hand, but the logic — every tolerance, however small, can be met — is the soul of the definition.
Step 1: plugging in gives 0/0 — inconclusive, not an answer.
Step 2: factor: (x² − 4)/(x − 2) = (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2.
Step 3: the limit only cares about x near 2, not at 2, so lim x→2 (x + 2) = 4.
The graph has a removable hole at (2, 4) — the limit is 4 even though f(2) is undefined.
2 · Left- and right-hand limits
lim x→a⁺ f(x) approaches from the right (x > a); lim x→a⁻ f(x) from the left (x < a). The two-sided limit exists if and only if both one-sided limits exist and agree. Piecewise functions are judged at their seams exactly this way.
Step 1: from the left: lim x→1⁻ (x + 1) = 2.
Step 2: from the right: lim x→1⁺ (4 − x) = 3.
Step 3: 2 ≠ 3, so the two-sided limit does not exist. The graph jumps from height 2 to height 3 at x = 1.
3 · When limits fail to exist
Three classic failures:
- Jump discontinuity — the one-sided limits exist but disagree (Example 2).
- Oscillation — f(x) = sin(1/x) near 0 swings between −1 and 1 infinitely often in any window around 0; it never settles, so no L exists.
- Blow-up — f(x) = 1/x² near 0 rockets past every number; the function grows without bound (we write lim x→0 1/x² = ∞ as shorthand, and ∞ is not a real number, so the limit fails to exist in the strict sense).
4 · Limits at ±∞ and asymptotes, correctly
lim x→∞ f(x) = L means f(x) settles near L as x grows without bound. For rational functions, compare degrees: same degree → ratio of leading coefficients; denominator higher degree → 0; numerator higher degree → unbounded.
Vertical asymptote at x = a: a one-sided (or two-sided) limit there is ±∞ — the function blows up near a. Horizontal asymptote y = L: lim x→∞ f(x) = L or lim x→−∞ f(x) = L — the function levels off far out. An asymptote is a statement about a limit, never about a single point, and graphs may cross horizontal asymptotes without invalidating them.
5 · The Sandwich Principle (squeeze theorem)
If g(x) ≤ f(x) ≤ h(x) near a (except possibly at a) and lim g = lim h = L, then lim f = L — the function is squeezed into L. This is how the course's most famous limit is proved.
Step 1: on the unit circle, for small x > 0, comparing areas gives sin x < x < tan x.
Step 2: divide by sin x: 1 < x/sin x < 1/cos x, so cos x < sin x / x < 1.
Step 3: both cos x and 1 squeeze to 1 as x → 0. By the Sandwich Principle, lim x→0 sin x / x = 1.
The same bound works for x < 0 by symmetry, since sin x / x is even. This limit is the engine behind every derivative of sin x, so it is worth seeing once.
More generally, if |f(x)| ≤ M·|x|^p near 0 with p > 0 you know instantly the limit is 0; and a function trapped between two identical limits has nowhere else to go.
lim x→0 sin x / x = 1 · lim x→0 (1 − cos x) / x = 0
Rational at ∞: same degree → ratio of leading coefficients · denominator higher → 0
Sandwich: g ≤ f ≤ h, lim g = lim h = L ⇒ lim f = L
ε–δ: ∀ ε > 0 ∃ δ > 0 with 0 < |x − a| < δ ⇒ |f(x) − L| < ε
Common mistakes
- Plugging in, getting 0/0, and writing "the limit is 0/0" — 0/0 means keep working.
- Assuming the two-sided limit exists because one side exists — both sides must exist and agree.
- Calling ∞ a value of the limit; it signals unbounded behavior, not a real-number answer.
- Confusing horizontal asymptotes with hard ceilings — graphs can cross them.
- Declaring lim x→0 sin(1/x) = 0 because sin equals 0 infinitely often; it never settles.
- Using L'Hôpital's rule before its preconditions (differentiable pieces, genuine 0/0 or ∞/∞) are met.