公式内容

$$\begin{aligned} & 1\lim_{x\to0}\frac{\sin x}{x}=1,一\text{般式:}\lim_{\Delta\to0}\frac{\sin\Delta}{\Delta}=1\\ & 2\lim_{x\to\infty}\biggl(1+\frac{1}{x}\biggr)^{x}=\lim_{x\to0}\bigl(1+x\bigr)^{\frac{1}{x}}=e,一\text{般式:}\lim_{\Delta\to0}\bigl(1+\Delta\bigr)^{\frac{1}{\Delta}}=e\end{aligned}$$

$x\rightarrow0时,x=\sin x=\tan x$

one

The World Pipe Band Championships will take place in Glasgow on Saturday. Over 120 of the world’s best pipe bands will compete at the championships. Over 4,000 pipers and drummers are expected to descend on the city for the annual event. It will take place at the Saint James Playing Fields in Glasgow on Saturday, 18 May from 10:00 am until 7:00 pm. The event is free. Also, a free shuttle bus will run between the event car parks and the competition site.Shuttle buses will run every 15 minutes from 8:00 am until 8:00 pm.[2]Glasgow City Council says it is estimated to bring 300,000 pounds to the local economy. The first World Pipe Band Championships were held in Edinburgh in 1947.The event first took place in Glasgow in 1948. It has been held regularly in Glasgow since 1986.

two

Summer doesn’t begin for two more days, but scientists are already sounding the alarm about dangerous ice melts going on right now in Greenland. The vast island usually locks away enough fresh water in its ice sheet, which if melted, can raise global sea levels by 6 metres. And research shows that in recent years, its ice has melted faster than ever before. This month, as temperatures in northwestern Greenland approached all-time highs, researchers are encountering unusual, surprising levels of melting ice in the region. The most significant concern here is how early this is all happening. [4] Greenland experiences annual ice melt and growth cycles. But this year, the melt ing season began in early May. That’s a month earlier than normal and something that’s happened only once before in the historical record in 2016. That means that not only is the ice melting very fast, it has more time to melt this summer than it would in a normal climate system. Longer melt seasons like this one seem to lead to much greater contributions to sea level rise than in previous decades.

three

Bundles containing thousands of pounds in cash keep turning up on the streets of Blackhall Colliery in northern England, according to police. Residents of the village have been coming across bundles of £20 notes, usually worth £2,000, in the former mining village in County Durham. The latest discovery was on Monday, meaning four of these cash bundles have been handed in this year and t hirteen since 2014. The bundles are usually left in plain sight and had been scattered across the small coastal village. [6]The owner has two weeks to collect the cash, which is then returned to the finder if not claimed, according to Peterlee Police. A Durham police spokesperson said the money has been returned to the lucky finders, in all cases, except yesterday. Policeman John Foster said in a statement:“The circumstances remain a mystery so we would welcome any information that will help us get to the bottom of these random incidents.”[7] He told The Northern Echo,“it isn’t an affluent part of the world, so for them to find cash of that amount and instantly think about taking it to the police stations shows the community spirit there is.”

four

W: It’s your birthday next week. What have you got planned?
M: I’m not sure. [8]I often feel strange on my birthday. It’s like my brain decides to have a crisis.
W: It’s not entirely unusual to feel anxious or sad around your birthday. Birthdays can tap into a lot of things people worry about, including their achievements in life, in the past decade or their accomplishments of the past year.[9]Many begin to search for the meaning of their existence, leading to behaviors such as ending or starting a relationship or plans like starting a vigorous diet or fitness program.
M: I have wonderful friends.[10] And I love celebrating their birthdays, but I don’t like being the center of attention,receiving gifts and having a fuss made. It seems to trigger a type of social anxiety. I think social media, too, can intensify things as birthdays now play out more publicly. Birthday reminders can be helpful, but I kind of worry Iwon’t be able to drum up my own day and show it to look as exciting as it is supposed to be.
W: Well, to deal with the birthday blues, you should not isolate yourself. It’s best to gradually face your birthday with people you trust. That would help you learn self-acceptance or that it’s okay to be the focus.
M: Perhaps. Or I could perceive my birthday as an opportunity to do something for others like asking people to make a donation instead of buying a gift.
W: Exactly.[11]Or even simply see your birthday as an opportunity to bring people together for them to have fun.

five

M: [15] Not if you use a ride sharing app.
W: [15] Good idea. I’ll download one immediately. Thank you.

感受极限

对于$\text{y=x}$，随着$\text{X}$的无限增大，$\text{y}$的值也会无穷增大
对于$y=x^{2}$，随着$\text{X}$的无限增大，$\text{y}$的值也会无穷增大
对于$\mathbf{y}=\frac1x$，随着$\text{X}$的无限增大，$\text{y}$的值无限接近于0
对于$\mathbf{y}={\frac{\mathbf{x}}{\mathbf{x}+1}}$，随着$\text{X}$的无限增大，$\text{y}$的值无限接近于1

对于$\text{y=x}$，$y=x^{2}$和$\mathbf{y}=\frac1x$的极限结果，都比较好理解
我们来看$\mathbf{y}={\frac{\mathbf{x}}{\mathbf{x}+1}}$是如何计算的
首先在$\text{X}$无限大时$\text{x+1}$也是无限大的，两者都是$+\infty $，
因此在$\text{X}$无限大时，$\text{x=x+1}$是成立的，分子分母相同，求出来就是1（其实这么做不是正确的，因为你不可能写一大段文字描述而且写文字描述也很不数学）

新东西

新定义

习惯于使用大学数学表示方法表示新定义，可以规范自己的语言

定义域

函数定义：设$D$ 是一个实数集合，对每一个$x\in D$,存在一个对应法则$f$,都能对应唯
一的一个实数 $y$,则这个对应法则$f$称为定义在 $D$ 上的一个函数，记为：$y=f(x)$

有界性

$$\begin{aligned}(1)&\text{有界性:}\quad\forall x\in D\text{,若存在正数}M,\text{ 都有}\mid f(x)\mid\leq M\text{ 成立,则称 }f(x)\text{ 在区间 }D\text{ 上}\\\text{有界。}\end{aligned}$$

反函数（逆映射）

我的通俗理解：
想象我们有一个函数 $f(x)$，它在坐标系中画出了一条曲线。现在，我们要找到这个函数的反函数 $f^{-1}(x)$。