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Master Limits and Continuity: 70 Flashcards From Epsilon-Delta to Cauchy Sequences

Study limits with 70 real flashcards: epsilon-delta, squeeze theorem, sin(x)/x, Cauchy sequences, two cards to rewrite and the continuity cards to add.

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Deck CC9636, Limits & Continuity, has 70 cards — and almost all of them are about limits. The deck is a card-by-card walk through the Wikipedia article "Limit (mathematics)": strong on definitions, sequences and series, then drifting into topology, non-standard analysis and dynamical systems. Continuity itself gets three cards. Here is how to study what's there and fill what isn't.

Study this deck: Limits & Continuity

1 / 10
Question

Limit in mathematics

Knew itLearning

Tap the card for the answer · swipe right if you knew it, left if not

Cards worth knowing cold

FrontBack
(ε, δ)-definition of a limitFor every ε > 0, there exists δ > 0 such that if 0 < |x − c| < δ, then |f(x) − L| < ε
Lim(x→1) (x² − 1)/(x − 1)2 (f(1) is undefined, but the expression simplifies to x + 1)
Lim(x→0) sin(x)/x1
Lim(x→∞) (1 + 1/x)^xe ≈ 2.71828
Squeeze theoremIf aₙ ≤ bₙ ≤ cₙ and lim aₙ = lim cₙ = L, then lim bₙ = L
At x = 0, left and right limits of I⁺(x)?Left = 0, right = 1, so the limit does not exist
Limit of 0.9, 0.99, 0.999, …1 (why 0.999… = 1)
Is ℚ complete?No — decimal approximations of √2 are Cauchy in ℚ but don't converge in ℚ

What the 70 cards cover

Limit notation and the ε-δ definition, one-sided limits, infinite limits and limits at infinity, the indeterminate form 0/0, limit laws for sums, products and quotients, the squeeze theorem, the derivative as a limit (two forms), convergent, divergent and oscillatory sequences, Cauchy sequences and complete metric spaces, partial sums, the Basel problem (Σ1/n² = π²/6), absolute vs conditional convergence, the Riemann series theorem, power series and radius of convergence, the ratio test, pointwise vs uniform convergence, order of convergence, limit points and closed sets, Hausdorff spaces, ω- and α-limit sets, the standard part function.

Study it in two layers

Layer 1 — Calculus I (about 20 cards): 1–3, 15–19, 35–36, 39–42, 50–52, 63–64, 69–70. This is the exam material. Only around ten of these actually compute something, so when you review card 17 or 50, work it on paper before flipping.

Layer 2 — analysis and beyond: Cauchy sequences, uniform convergence, series theorems, then the topology, recursion-theory and dynamical-systems cards (37–38, 43–48, 55–61, 65–66). Suspend the second group if your goal is an AP or first-year exam.

Two cards to rewrite

  • Card 10 says a bounded sequence is "a sequence that does not tend to infinity". False: aₙ = (−1)ⁿ·n never tends to ±∞ yet is unbounded. Correct back: there is M with |aₙ| ≤ M for all n.
  • Card 54 makes the mirror mistake ("unbounded = tends to ±∞"). Same counterexample; tending to infinity implies unbounded, not the reverse.

Write the continuity cards yourself

The title promises continuity; add these before you start:

  1. f is continuous at c → lim(x→c) f(x) = f(c) (three conditions: f(c) defined, limit exists, they're equal).
  2. Removable / jump / infinite discontinuity → one card each, with (x² − 1)/(x − 1), I⁺(x) and 1/x as the deck's own examples.
  3. Intermediate Value Theorem and Extreme Value Theorem.
  4. L'Hôpital's rule — card 52 names it but never states it.

Find the deck in the Flip It library with code CC9636. Once limits are solid, continue with derivatives and series and sequences.

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