函数极限定义

自变量变化的描述:

  1. x→x0:∃δ>0x \rightarrow x_0 : \exists \delta > 0, 当0<∣x−x0∣<δ0<|x-x_0|<\delta,
  2. x→x0+:∃δ>0x \rightarrow x_0^+ : \exists \delta > 0, 当0<x−x0<δ0<x-x_0<\delta,
  3. x→x0−:∃δ>0x \rightarrow x_0^- : \exists \delta > 0, 当0<x0−x<δ0<x_0-x<\delta,
  4. x→∞:∃X>0x \rightarrow \infty : \exists X > 0, 当∣x∣>X|x|>X,
  5. x→+∞:∃X>0x \rightarrow +\infty: \exists X > 0, 当x>Xx>X,
  6. x→−∞:∃X>0x \rightarrow -\infty: \exists X > 0, 当x<−Xx<-X,

因变量变化的描述:

  1. f(x)→A:∀ε>0,∃...f(x) \rightarrow A : \forall \varepsilon > 0, \exists...,当..., ∣f(x)−A∣<ε|f(x) - A| < \varepsilon
  2. f(x)→∞:∀M>0,∃...f(x) \rightarrow \infty : \forall M > 0, \exists...,当..., ∣f(x)∣>M|f(x)| > M
  3. f(x)→+∞:∀M>0,∃...f(x) \rightarrow +\infty : \forall M > 0, \exists...,当..., f(x)>Mf(x) > M
  4. f(x)→−∞:∀M>0,∃...f(x) \rightarrow -\infty : \forall M > 0, \exists...,当..., f(x)<−Mf(x) < -M

例子:

  1. lim⁡x→∞f(x)=+∞⇔∀M>0,∃X>0\lim\limits_{x\rightarrow{\infty}}f(x)=+\infty \Leftrightarrow \forall M>0, \exists X>0,当∣x∣>X|x|>X时,有f(x)>Mf(x)>M
  2. lim⁡x→∞f(x)=A⇔∀ε>0,∃X>0,\lim\limits_{x\rightarrow\infty} f(x)=A \Leftrightarrow \forall \varepsilon>0, \exists X>0, 当∣x∣>X|x|>X时, 有∣f(x)−A∣<ε|f(x)-A|<\varepsilon
  3. lim⁡x→x0f(x)=+∞⇔∀M>0,∃δ>0\lim\limits_{x\rightarrow{x_0}}f(x)=+\infty \Leftrightarrow \forall M>0, \exists \delta>0,当0<∣x−x0∣<δ0<|x-x_0|<\delta时,有∣f(x)∣>M|f(x)|>M
  4. lim⁡x→x0f(x)=A⇔∀ε>0,∃δ>0,\lim\limits_{x\rightarrow x_0} f(x)=A \Leftrightarrow \forall \varepsilon>0, \exists\delta>0, 当0<∣x−x0∣0<|x-x_0|时, 有∣f(x)−A∣<ε|f(x)-A|<\varepsilon