Limits of Real Functions: Examples and Counterexamples

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1 · Prerequisites

Real Analysis

Set Theory

2 · Logical flowchart

3 · Items

Remark

The classical form of the oscillator above is sin(1/x), which this library can only construct much later

Example

Every polynomial has lim_x → c p(x) = p(c), and rational functions do so away from the zeros of the denominator

Example

The trigonometry-free oscillator ψ(x) = inf_n ∈ ℤ |x - n| is well defined and attained at a nearest integer, takes values in [0, 1/2], vanishes exactly on ℤ, equals 1/2 at half-integers, and is 1-periodic

Example

Limits of Real Functions: Examples and Counterexamples

3 · Items

The classical form of the oscillator above is sin(1/x), which this library can only construct much later

Every polynomial has lim_x → c p(x) = p(c), and rational functions do so away from the zeros of the denominator

The trigonometry-free oscillator ψ(x) = inf_n ∈ ℤ |x - n| is well defined and attained at a nearest integer, takes values in [0, 1/2], vanishes exactly on ℤ, equals 1/2 at half-integers, and is 1-periodic

x ψ(1/x) → 0 as x → 0, by the squeeze theorem

The sign function has both one-sided limits at 0 and no two-sided limit

(3x² - 1)/(x² + x) → 3 as x → +∞

ψ(1/x) has no limit at 0: two sequences tending to 0 give values constantly 0 and constantly 1/2

The function equal to 0 off the origin and to 1 at the origin has limit 0 ≠ 1 there

With g ≡ 0 and f equal to 0 off the origin and 1 at it, lim g = 0 and lim_y → 0 f = 0 while f ∘ g ≡ 1

The indicator of ℚ has a limit at no point of ℝ

x · 1_ℚ(x) has a limit at 0 and at no other point

On the domain {0} ∪ [1,2] every real is vacuously a limit at 0