5. LIMITS OF FUNCTIONS AND CONTINUITY

Limit of a function of one real variable as x → x₀

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A. The function converges to lim_x→x0 f(x) = L ∈ ℝ if, for every ε > 0, there exists δ_ε > 0 such that, for every x ∈ A, 0 < |x - x₀| < δ_ε → |f(x) - L| < ε

Donut neighborhood

The donut neighborhood of center x₀ and radius δ > 0 is a neighborhood of x₀ deprived of x₀ itself: B_δ(x₀) -{x₀} = (x₀ - δ, x₀) ∪ (x₀, x₀ + δ)

For a function f: A ⊆ ℝ → ℝ: x ∈ B_δ(x₀) -{x₀} → |f(x) - L| < ε, for every x ∈ A

Limit of a function of one real variable as x → x₀ using neighborhoods

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A. The function converges to lim_x→x0 f(x) = L ∈ ℝ if, for every B_ε(L) of L, there exists U_δε(x₀) such that x ∈ U_δε(x₀)∩A → f(x) ∈ B_ε(L), where x ≠ x₀

Limit of a function of one real variable as x → x₀ ∈ Ṝ using neighborhoods

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ Ṝ be a limit point of A. The function converges to lim_x→x0 f(x) = L ∈ Ṝ if, for every B(L) of L, there exists U(x₀) such that x ∈ U(x₀)∩A → f(x) ∈ B(L), where x ≠ x₀

Positive infinite limit of a function of one real variable as x → x₀

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A. The function converges to lim_x→x0 f(x) = +∞ if, for every K > 0, there exists δ_K > 0 such that, for every x ∈ A, 0 < |x - x₀| < δ_K → f(x) > K

Limit of a function of one real variable as x → +∞

Given a function f: A ⊆ ℝ, → ℝ, let A be unbounded above. The function converges to lim_{x→+ ∞} f(x) = L ∈ ℝ if, for every ε > 0, there exists M_ε > 0 such that, for every x ∈ A, x > M_ε → |f(x) - L| < ε

Positive infinite limit of a function of one real variable as x → ∞

Given a function f: A ⊆ ℝ, → ℝ, let A be unbounded above. The function converges to lim_x→+∞ f(x) = +∞ if, for every K > 0, there exists M > 0 such that, for every x ∈ A, x > M → f(x) > K

Limit in Ṝ from the right of a function of one real variable as x → x₀⁺

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A∩ [x₀, +∞). The right limit at x₀ lim_x→x0+ f(x) = L ∈ Ṝ if, for every B(L) of L, there exists U_δε⁺(x₀) = [x₀, x₀ + δ_ε) such that x ∈ U_δε⁺(x₀)∩A → f(x) ∈ B(L), where x ≠ x₀

Limit in ℝ from the right of a function of one real variable as x → x₀⁺

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A∩ [x₀, +∞). The right limit at x₀ lim_x→x0+ f(x) = L ∈ ℝ for every ε > 0, there exists δ_ε > 0 such that, for every x ∈ A, x₀ < x < δ_ε + x₀ → |f(x) - L| < ε

Positive right limit of a function of one real variable as x → x₀⁺

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A∩ [x₀, +∞). The right limit at x₀ lim_x→x0+ f(x) = +∞ if, for every K > 0, there exists δ_K > 0 such that, for every x ∈ A, x₀ < x < δ_K + x₀ → f(x) > K

Connection between limits and one-sided limits

The limit of a function at a point x₀ exists if and only if its one-sided limits at that point are equal

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A∩ [x₀, +∞) and (-∞, x₀]∩A:

lim_x→x0 f(x) = L ∈ Ṝ ←→ lim_x→x0+ f(x) = lim_x→x0- f(x) = L ∈ Ṝ

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ Ṝ be a limit point of A. The function converges to lim_x→x0 f(x) = L ∈ Ṝ if, for every neighborhood B(L), there exists a neighborhood U of x₀ such that f((U\{x₀}) ∩ A) ⊆ B(L)

Vertical asymptote

A vertical asymptote is line x = x₀ if either lim_x→x0+ f(x) = +-∞ or lim_x→x0- f(x) = +-∞

Horizontal asymptote

A horizontal asymptote is a line y = L ∈ ℝ, if either lim_x→+∞ f(x) = L, or lim_x→-∞ f(x) = L

Given a function f: A ⊆ ℝ, → ℝ let x₀ ∈ ℝ be a limit point of A. The function converges to lim_x→x0 f(x) = L ∈ Ṝ if, for every neighborhood B(L), there exists a neighborhood U_δ of x₀ such that U_δ(x₀) ∩ A) → f(x) ∈ B(L), where x ≠ x₀

Let f : ℝⁿ, → ℝ be given by f(x₁,x₂) = log(1 + x₁x₂)/x₁² . The limit of f does as (x1,x2) approaches (0,0) does not exist

Characterization of limits with limits of sequences

Given a function f: A ⊆ ℝⁿ, → ℝ let x₀ ∈ A’. The limit lim_x→x0 = L ∈ Ṝ if and only if x_n → x₀ → f(x_n) → L, for every {x_n} ⊆ A \ {x₀}

Theorem on the uniqueness of the finite limit

Let f: A ⊆ ℝⁿ, → ℝ and x₀ ∈ A’. There exists at most one L ∈ ℝ such that lim_x→x0 f(x) = L

By contradiction, suppose L’, L’’ are two different limits of f(x) as x→ x₀. Let {x_n} be a sequence whose terms belong to A, where eventually x≠ x₀ such that x → x₀

Then f(x_n) → L’ and f(x_n) → L’’, which contradicts the uniqueness of the limit for sequences. Therefore, the limit is unique

Comparison criterion for limits

Let f, g, h : A ⊆ ℝⁿ, → ℝ and x₀ ∈ A’. If g(x) ≤ f(x) ≤ h(x) for every x ∈ A, and lim_x→x0 g(x) = lim_x→x0 h(x) = L ∈ Ṝ, then lim_x→x0 f(x) = L