Science Metro

実関数の収束

$近傍$

a

の

ε

近傍

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

$n次元実数列の極限$

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

\

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

$実関数の極限$

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)

$左・右極限$

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 ∣)

$極限の四則演算$

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 ∣

$はさみうちの原理$

\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

$合成関数の極限$

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)

$n次元実数列$

n

次元実数列

: R^{n}

の点列

$命題$

α = (α_{1}, ..., α_{n})

のとき

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}

$領域$

s

$2変数関数$

2

変数関数

:

写像

f : D \to R

f

の定義域

: D

<br>

f

の値域

: f (D)

$2変数関数の極限$

(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)

$2変数関数の極限の演算$

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 ∣

$はさみうちの定理$

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