Function Limit

极限定义

$\lim_{x \to a}f(x)=L \Leftrightarrow \lim_{x \to a^{-}}f(x)=L$ 且 $\lim_{x \to a^{+}}f(x)=L$

The Epsilon-Delta Definition of a Limit

The limit of a function $f(x)$ as $x$ approaches a value $c$ is a number $L$ , written as $\lim_{x \to c} f(x) = L$ , if for every number $\epsilon > 0$ , there exists a number $\delta > 0$ such that if $0 < |x - c| < \delta$ , then $|f(x) - L| < \epsilon$ .

Conditions of continuous

$f(a)$ 存在，函数在 $x = a$ 处有定义
$\lim_{x \to a}f(x)$ 存在，函数在 $x = a$ 处有极限
$\lim_{x \to a}f(x)=f(a)$ ，函数在 $x = a$ 处的极限值等于函数在该点的值

运算法则

[!note] Proposition 3.1 If the functions $f$ and $g$ are continuous at $a$ , then

$f \pm g$ is continuous at $a$

$fg$ is continuous at $a$

$\frac{f}{g}$ is continuous at $a$ , provided $g(a) \ne 0$

$\lim_{x \to a}[kf(x)\pm g(x)] = k\lim_{x \to a}f(x)\pm\lim_{x \to a}g(x)$
$\lim_{x \to a}[f(x)\cdot g(x)]=\lim_{x \to a}f(x)\cdot\lim_{x \to a}g(x)$
$\lim_{x \to a}\frac{f(x)}{g(x)}=\frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}=\frac{A}{B}(B\neq0)$

拆极限

存在 + 存在 = 存在
不存在 + 存在 = 不存在
不存在 + 不存在 = 不一定存在

Pinching Theorem

Theorem 2.1 (Infinity)

Suppose that $f$ , $g$ and $h$ are all defined on the interval $(b, \infty)$ , where $b \in \mathbb{R}$ .
$f(x) \leq h(x) \leq g(x), \quad b < x < \infty$
$\lim_{x \to \infty} f(x) = L = \lim_{x \to \infty} g(x)$
$\lim_{x \to \infty} h(x) = L$ .

Theorem 2.3 （At $x = a$ ) Suppose that $f, g$ and $h$ are all defined on an open interval, $I$ , containing the point $a$ ; $f, g$ and $h$ may or may not be defined at $x=a$ .

$f(x) \le h(x) \le g(x) \quad \text{for } x \in I, x \ne a \quad \text{and}$

$\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = L \quad \text{then}$

$\lim_{x \to a} h(x) = L.$

All limit rules and the pinching theorem so far apply equally to the five forms of limit: $\lim_{x \to -\infty} f(x), \quad \lim_{x \to a^-} f(x), \quad \lim_{x \to a} f(x), \quad \lim_{x \to a^+} f(x), \quad \lim_{x \to \infty} f(x)$

Limits and composition of functions

Proposition 2.2 If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to L} g(x) = k$ and $g(L) = k$ , then we have $\lim_{x \to a} (g \circ f)(x) = \lim_{x \to a} g(f(x)) = g(\lim_{x \to a} f(x)) = g(L) = k.$ Here we can change $\lim_{x \to a}$ but cannot change conditions on $g$ . Under these conditions we say that $g$ is continuous at $L$ .

Continuity at a point

Definition 2.3 Suppose that $f$ is ==defined== on some ==open interval== containing the point $a$ . If $\lim_{x \to a} f(x) = f(a),$ we say that $f$ is continuous at $a$ ; otherwise we say that $f$ is discontinuous at $a$ .

Note: $f$ is ==not== continuous at $a$ if either

$f$ is not defined at $a$ ; or
$f$ is defined at $a$ but $\lim_{x \to a} f(x)$ does not exist; or
$f$ is defined at $a$ and $\lim_{x \to a} f(x)$ exists but $\lim_{x \to a} f(x) \ne f(a)$ .

The Intermediate Value Theorem

Theorem 3.1 Suppose that

$f$ is continuous on $[a, b]$ , and
$f(a) \ne f(b)$ . Then for any $d$ lying between $f(a)$ and $f(b)$ , there exists at least one point $c \in (a, b)$ satisfying $f(c) = d$ .

Applications of the IVT

Often, the IVT is used to show that the equation $f(x) = 0$ has a solution in $(a, b)$ .

In this case you need to check:

$f$ is continuous on $[a, b]$ , and
$f(a)f(b) < 0$ .

(i.e. $f(a)$ and $f(b)$ are the opposite sign of each other - meaning that to get from $f(a)$ to $f(b)$ you have to cross through zero.)

七种未定式（不能直接带入求值）

$(\frac{0}{0},\frac{\infty}{\infty},0\cdot\infty,\infty - \infty,1^{\infty},\infty^{0},0^{0})$

等价无穷小（使用时将 $x$ 广义化为趋于 0 的函数）

$\sin x \sim x$ , $\tan x \sim x$ , $\arcsin x \sim x$ , $\arctan x \sim x$ , $e^{x}-1 \sim x$ , $\ln(1 + x) \sim x$
$a^{x}-1 \sim x\ln a$ , $(1 + x)^{\alpha}-1 \sim \alpha x$
$1 - \cos x \sim \frac{1}{2}x^{2}$ , $x - \ln(1 + x) \sim \frac{1}{2}x^{2}$
$x - \sin x \sim \frac{1}{6}x^{3}$ , $x - \arcsin x \sim -\frac{1}{6}x^{3}$ , $x - \tan x \sim -\frac{1}{3}x^{3}$ , $x - \arctan x \sim \frac{1}{3}x^{3}$

只有整个函数的乘除才能用等价无穷小

洛必达法则

$x \to a$ 时， $f(x)$ 与 $g(x)$ 都趋于 0 或 $\infty$ （ $x \to \infty$ 也适用）
$f'(x)$ 与 $g'(x)$ 存在， $g'(x)\neq0$
$\lim_{x \to a}\frac{f'(x)}{g'(x)}$ 存在，或为 $\pm\infty$ ， $\lim_{x \to a}\frac{f(x)}{g(x)}=\lim_{x \to a}\frac{f'(x)}{g'(x)}$

重要极限

$\lim_{x \to \infty}(1 + \frac{1}{x})^{x}=e$
$\lim_{x \to 0}(1 + x)^{\frac{1}{x}}=e$
$\lim_{\square \to \infty}(1 + \frac{1}{\square})^{\square}=e$

幂指函数极限（ $1^{\infty},\infty^{0},0^0$ ）

$\lim u^{v}=\lim e^{v\ln u}=e^{\lim v\ln u}=e^{\lim v(u - 1)}$ （当 $u \to 1$ 时， $\ln u=\ln[1+(u - 1)]\sim u - 1$ ）