Chapter -- Limits 极限

Posted 2020-08-08

1. 曲线的切线问题和速度问题

 某一个点的切线是在这个点的基础上，一条与曲线相切的线，同时，切线的方向应该和曲线的方向是一致的。但是如果这条“切线”和曲线有了不止一个的切点。这个问题就不能这样考虑了。

因此，我们可以利用极限或者无限接近极限的思路来考虑这个问题。

 

2. 函数的极限

ƒ(x)的极限，随着x趋近于a，等于L

The left-hand/right-hand limit of ƒ(x) as x approaches a equals L;

The limit of ƒ(x) as x approaches a from the left/right is equal to L.

如果左极限和右极限不同的时候，该函数的极限是不存在的。

如果x趋近于a，函数的极限为无穷大。这里的∞并不是在表达数字或者任何数学的意义，它同样也是在说明该函数的极限是不存在的，只不过是换了一种说法而已。

 

3. 如何用极限的运算法则来计算极限

lim[ƒ(x) + g(x)] = limƒ(x) + limg(x)

lim[ƒ(x) – g(x)] = limƒ(x) – limg(x)

lim[C*ƒ(x)] = C*limƒ(x)

lim[(ƒ(x)*g(x)] = limƒ(x)*limg(x)

lim[ƒ(x)/g(x)] = limƒ(x)/limg(x) if limg(x)≠0

lim[ƒ(x)]ⁿ = [limƒ(x)]ⁿ

 

4. 连续

The limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. Notice that definition implicitly requires three things if ƒ is continuous at a:

1. ƒ(a) is defined.

2. limƒ(x) is existed.

3. limƒ(x) = ƒ(a) 

A continuous function ƒ has the property that a samll change in x produces only a samll change in ƒ(x). 一个连续的函数，当自变量x的有很小的变化的时候，因变量y也会相对应地产生很小的变化。

除此之外，我们都说这个函数是个不连续的函数。We say that ƒ is discontinuout at a.

 

定理1：

If ƒ and g are continuous at a and c is constant, then the following functions are also continuout at a:

ƒ+g;

ƒ–g;

cƒ;

ƒ*g

ƒ/g

 

定理2：

Any polynomial is continuous everywhere, that is, it is continuous on (-∞, +∞).

Any rational function is continuous wherever it is defined, that is, it is continuous on its domain.

 

定理3：

The polynomial/rational functions/root functions/trigonometric functions are continuous at every number in their domain.

多项式，除函数，平方根函数和三角函数在他们定义域上的每一个值上都是连续的。

 

定理4：

If ƒ is continuous at b and lim_x→ag(x)=b, then lim_x→aƒ(g(x)) = ƒ(b), that is:

lim_x→aƒ(g(x)) = ƒ(lim_x→a g(x))

 

定理5：

If g is continuous at a and ƒ is continuous at g(a), then the composite function ƒ°g given by (ƒºg)(x) is continuous at a.

 

The Intermediate value theorem:

Suppose that ƒ is continuous on the closed interval [a,b] and let N be any number between ƒ(a) and ƒ(b), where ƒ(a)≠ƒ(b). Then there exists a number c in (a,b) such that ƒ(c) = N. Note that value N can be taken on once or more than once. 

连续性函数就是X-Y轴上连续的，没有洞的函数。X轴上的一个点c，一定会对应Y轴上一个或多个的ƒ(c)的。