Part 1.极限：
一、定义：
<一> 极限
①\(\begin{align} & (\epsilon \to N) If \forall \epsilon > 0, \exists N > 0 当 n > N时 \\ &|a_n – A| < \epsilon \\ & \Rightarrow \lim_{n \to \infty} a_n = A \end{align}\)
②\(\begin{align} &(\epsilon \to \delta)If \forall \epsilon > 0, \exists \delta > 0 , 当 0 < |x-a| < \delta时,\\ &|f(x) – A| < \epsilon \\ & \Rightarrow \lim_{x \to \infty} f(x) = A \end{align}\)
③\(\begin{align} &(\epsilon \to x) \begin{cases} x \to -\infty \\ x \to +\infty \\ x \to \infty \end{cases} \\ & If \forall \epsilon >0 , \exists X>0,当x > X时,\\&|f(x) -A |< \epsilon \\ &\Rightarrow \lim_{x \to \infty} f(x) = A\end{align}\)
<二> 无穷小
\( If \lim_{x \to a} \alpha(x) = 0 ,称 \alpha(x)当 x \to a 时为无穷小\)\(设\alpha \to 0, \beta \to 0 \begin{cases} If \lim \frac{\beta}{\alpha} = 0 \rightarrow \beta = o(\alpha) \\ If \lim \frac{\beta}{\alpha} = k (k \neq 0, k \neq \infty) \rightarrow \beta = O(\alpha) \\ If \lim \frac{\beta}{\alpha} = 1 \rightarrow \alpha \to \beta \end{cases}\)
二、性质
<一>一般性质：
1、（唯一性）
2、（保号性）
\(\begin{align} &\lim_{x \to a} f(x) = A \begin{cases} >0 \\ <0\end{cases} \rightarrow \\ & \exists \delta > 0,当 0 < |x-a| < \delta 时 \\ & f(x) \begin{cases} > 0 \\ < 0 \end{cases}\end{align}\)3、（有界性）
\(\begin{align}& ① \lim_{n \to \infty} a_n = A \to \exists M > 0 使 |a_n| \leq M ,（反之不对）\\&② \lim_{x \to a}f(x) = A \rightarrow \exists \delta > 0, M > 0 \\ &当 0 < |x -a| < \delta 时, |f(x)| \leq M (局部有界性)\end{align}\)<二>运算性质
1、\( \lim f(x) =A,\lim g(x) = B 则 \begin{cases}①\lim[f(x) \pm g(x)] = A \pm B \\ ② \lim[f(x)g(x)] = AB \\ ③\lim[f(x)/g(x)] = \frac{A}{B} (B \neq 0) \end{cases}\)
2、\(\begin{align} &\lim_{u \to a} f(u) = A,u = \varphi (x)且 \varphi(x) \neq a \\&则 \lim_{x \to x_0} \varphi (x) = a, 则 \lim_{x \to x_0}f[\varphi (x)] = A\end{align}\)
<三>存在性质
1、夹逼准则<迫敛定理> \(\begin{align}If \begin{cases} a_n \leq b_n \leq c_n \\ \lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = A \end{cases} \Rightarrow \lim_{n \to \infty }b_n = A \end{align}\)
2、单调有界的数列必定存在极限
<四>无穷小性质
1、一般性质
\(① \alpha \to 0 , \beta \to 0 \Rightarrow \begin{cases} \alpha \pm \beta \to 0 \\ \alpha \beta \to 0 \\ \beta \alpha \to 0\end{cases}\)
\(②|\alpha| \leq M , \beta \to 0 \Rightarrow \alpha \beta \to 0 \)
\(③\lim f(x) = A \Leftrightarrow f(x) = A + \alpha, \alpha \to 0\)