定义 \begin{aligned} &\lim_{x\to\cdot}f(x)=A\iff\\ &\forall\varepsilon>0,x\to\cdot时,|f(x)-A|0,0x\end{cases}\\ \end{aligned} 性质唯一性 \begin{aligned} &A唯一：左极限、右极限；左导、右导\\ \color{maroon}[例]&求\lim_{x\to0}\frac{\tan\pi x}{|x|(x^2-1)}\\ &\color{black}I_+=\lim_{x\to0^+}\frac{\tan\pi x}{x(x^2-1)}=\lim_{x\to0^+}\frac{\pi x}{x(-1)}=-\pi\\ &I_-=\lim_{x\to0^-}\frac{\pi x}{(-x)(-1)}=\pi\\ &\implies I不\exists\\ \color{grey}[注]&如|x|,e^x,\arctan x需要考虑这种情况 \end{aligned} A是一个数 \begin{aligned} &A是一个数，记\lim_{x\to\cdot}f(x)=A\\ \color{maroon}[例]&已知\lim_{x\to1}f(x)存在，f(x)=\frac{x-\arctan(x-1)-1}{(x-1)^3}+2x^2e^{x-1}\cdot\lim_{x\to1}f(x)\\ &\color{black}\lim_{x\to1}f(x)=\lim_{x\to1}\frac{(x-1)-\arctan(x-1)}{(x-1)^3}+A\lim_{x\to1}2x^2e^{x-1}\\ &\implies A=\lim_{t\to0}\frac{t-\arctan t}{t^3}+2A\\ &\implies A=-\frac13\\ &\therefore f(x)=\frac{x-\arctan(x-1)-1}{(x-1)^3}-\frac23x^2e^{x-1}\\ \color{grey}[注]&以后会知道，"只要存在"，则有\lim_{x\to\cdot}f(x)=A,\lim_{x\to\infty}x_n=A\\ &f''(x_0)=A;\int_a^bf(x)dx=A;\iint_Df(x,y)d\sigma=A \end{aligned} 有界性 \begin{aligned} &x\to\cdot,|f(x)|\leq M\\ \color{maroon}[例]&若\lim_{x\to x_0}\frac{f(x)}{x-x_0}=A(存在),求\lim_{x\to x_0}f(x)\\ &\color{black}\lim_{x\to x_0}f(x)=\lim_{x\to x_0}\frac{f(x)}{x-x_0}\cdot(x-x_0)=0(前者有界函数，后者无穷小)\\ \color{grey}[注]&若增加“f(x)在x_0处连续”\implies f(x)=\lim_{x\to x_0}f(x)=0 \end{aligned} 局部保号性 \begin{aligned} &x\to\cdot,若A>0,\implies f(x)>0(局部保号)(不等式脱帽法)\\ \color{maroon}[例]&证明：当x\to0^+时，0