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Limits & Continuity

70 cards · 🇺🇸 English

Limits, one-sided limits, continuity and asymptotic behavior.

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Limit in mathematics

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1Limit in mathematics
The value that a function or sequence approaches as the argument or index approaches some value
2lim (x→c) f(x) = L
The limit of f(x) as x approaches c equals L; the value of f(x) can be made arbitrarily close to L by choosing x sufficiently close to c
3(ε, δ)-definition of a limit
For every ε > 0, there exists δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε
4Convergent sequence
A sequence that has a limit
5Divergent sequence
A sequence that does not have a limit
6Can a convergent sequence have more than one limit?
No, a convergent sequence has only one limit
7A sequence to 'tend to infinity'
For each real number M (bound), there exists an integer N such that for all n > N, aₙ > M
8Oscillatory sequence
A divergent sequence that does not tend to infinity (e.g., aₙ = (-1)ⁿ)
9Limit of the sequence 0.9, 0.99, 0.999, ...
1 (this explains why 0.999... = 1)
10Bounded sequence
A sequence that does not tend to infinity
11Pointwise convergence
A sequence of functions {fₙ} converges pointwise to f if for each x, lim(n→∞) fₙ(x) = f(x)
12Uniform convergence
A sequence of functions converges uniformly if the maximum difference between the functions approaches zero as n→∞
13What property does uniform convergence preserve that pointwise convergence does not?
Continuity (the uniform limit of continuous functions is continuous)
14What property do continuous functions have regarding limits?
They preserve limits: if lim aₙ = L, then lim f(aₙ) = f(L)
15Left-handed limit
The limit of f(x) as x approaches c from values less than c, denoted lim(x→c⁻) f(x)
16Right-handed limit
The limit of f(x) as x approaches c from values greater than c, denoted lim(x→c⁺) f(x)
17Lim(x→1) (x² - 1)/(x - 1)
2 (despite f(1) being undefined, the limit simplifies to x + 1, which equals 2 at x = 1)
18Lim(x→∞) (2x - 1)/x
2
19A limit to tend to infinity in the value
For any real M, there exists δ > 0 such that if 0 < |x - c| < δ, then |f(x)| > M
20Cauchy sequence
A sequence where for every ε > 0, there exists N such that for all m, n > N, |aₘ - aₙ| < ε
21Are all convergent sequences Cauchy sequences?
Yes, for sequences of real numbers
22Are all Cauchy sequences convergent?
For real numbers, yes. In general metric spaces, not necessarily
23Complete metric space
A metric space where every Cauchy sequence is convergent
24Is the set of rational numbers (ℚ) complete?
No (e.g., decimal approximations of √2 form a Cauchy sequence in ℚ but don't converge in ℚ)
25Limit set of a sequence
The set of all points that are limits of convergent subsequences
26Infinite series
An expression of the form Σ(n=1 to ∞) aₙ, defined as the limit of partial sums
27Partial sum
The sum of the first n terms: sₙ = a₁ + a₂ + ... + aₙ
28Basel problem result
Σ(n=1 to ∞) 1/n² = π²/6
29Unconditional convergence
A series that converges for all orderings of terms (equivalent to absolute convergence)
30Absolute convergence
A series Σaₙ is absolutely convergent if Σ|aₙ| converges