Convergence of real function

$Neighborhood$

ε

-neighborhood of

a

:

U_{ε} (a) = {x \in R^{n} ∣∣ x - a ∣∣ < ε}

n \to \infty lim a_{n} = a :

\forall ε > 0\exists N \in N (n \geq N \to ∥ a_{n} - a ∥ < ε)

x \to a lim f (x) = A :

\forall ε > 0\exists δ > 0, \forall x \in I (0 < ∣ x - a ∣ < δ \to ∣ f (x) - A ∣ < ε)

x \to a + 0 lim f (x) = A :

\forall ε > 0\exists δ > 0, \forall x \in I (0 < x - a < δ \to ∣ f (x) < ε ∣)

x \to a - 0 lim f (x) = A :

\forall ε > 0\exists δ > 0, \forall x \in I (0 < a - x < δ \to ∣ f (x) < ε ∣)

1. λ, μ \in R \Rightarrow x \to a lim {λ f (x) + μg (x)} = λ α + μ β 2. x \to a lim f (x) g (x) = α β 3. β \neq = 0 \Rightarrow x \to a lim \frac{f ( x )}{g ( x )} = \frac{α}{β} 4. x \to a lim ∣ f (x) ∣ = ∣ α ∣

\exists δ > 0, (f (x) \leq g (x) \leq h (x) (x \in U_{δ} (a), x \neq = a)

x \to a lim f (x) = x \to a lim g (x) = α \Rightarrow x \to a lim h (x) = α

x \to a lim f (x) = b, y \to b lim g (y) = g (b) \Rightarrow x \to a lim g (f (x)) = g (b)

n-dimensional real sequence: sequence in

R^{n}

n \to \infty lim a_{n} = α \Leftrightarrow n \to \infty lim a_{n_{1}} = α_{1}, ..., n \to \infty lim a_{n_{n}} = α_{n}

2-variable function:

f : D \to R

Domain of

f

: D

Range of

f

: f (D)

(x, y) \to (a, b) lim f (x, y) = α :

\forall ε > 0\exists δ > 0\forall x \in D (0 \neq = ∣∣ x - a ∣∣ < δ \to ∣ f (x, y) - α ∣ < ε)

1. λ, μ \in R \Rightarrow (x, y) \to (a, b) lim {λ f (x, y) + μg (x, y)} = λ α + μ β 2. (x, y) \to (a, b) lim f (x, y) g (x, y) = α β 3. β \neq = 0 \Rightarrow (x, y) \to (a, b) lim \frac{f ( x , y )}{g ( x , y )} = \frac{α}{β} 4. (x, y) \to (a, b) lim ∣ f (x, y) ∣ = ∣ α ∣

(x, y) \to (a, b) lim g (x, y) = 0 \Rightarrow (x, y) \to (a, b) lim f (x, y) = α