Let the function f(x) be defined in the centripetal neighborhood of point X. If there is a constant A, there is always a positive number δ for any given positive number ε (no matter how small it is), so that when x satisfies inequality 0.
| f(x)-A | & lt； ε
Then the constant a is called the time limit of the function f(x) when x → x
Popular definition of function limit;
1. Let the function y=f(x) be defined in (a, +∞). If the function f(x) is infinitely close to a constant a when x→+∞, then A is called the limit of the function f(x) when x tends to +∞. Let it be written as lim f (x) = a, x→+∞.
2. Let the function y=f(x) be defined near point A. When x approaches a infinitely (denoted as x→a), the value of the function approaches a constant infinitely, then A is called the limit of the function f(x) when x approaches a infinitely. Write lim f(x)=A, x → a.
Left and right limits of function:
1: if the function f(x) approaches the constant a infinitely when x approaches x0 from the left side of point x=x0 (that is, x < x0), it is said that a is the left limit of the function f(x) at point x0, and it is denoted as x→ x0-LIMF (x) = a. 。
2. if x is from the right side of point x=x0 (i.e. x >；; When x0) infinitely approaches the point x0, the function f(x) infinitely approaches the constant a, that is, A is the right limit of the function f(x) at the point x0, which is denoted as x→ x0+LIMF (x) = a. 。
Note: If the left and right limits of a function are different on x(0), then the function has no limit on x(0).
Note: whether the function has a limit at x(0) has nothing to do with whether it is defined at x=x(0), as long as y=f(x) is defined near x(0).
Properties of function limit:
Limit algorithm (or related formula):
lim(f(x)+g(x))=limf(x)+limg(x)
lim(f(x)-g(x))=limf(x)-limg(x)
lim(f(x)*g(x))=limf(x)*limg(x)
Lim (f (x)/g (x)) = LIMF (x)/LIMG (x) (LIMG (x) is not equal to 0).
lim(f(x))^n=(limf(x))^n
Only when the above limf(x) limg(x) exists can it be established.
lim( 1+ 1/x)^x =e
x→∞
Infinity and infinitesimal:
A series (limit) is infinitely close to 0, which is an infinitesimal series (limit).
Infinite sequence and infinitesimal sequence are reciprocal.
Two important limitations:
1、lim sin(x)/x = 1，x→0
2.lim (1+ 1/x) x = e, x→∞ (e≈2.7 1828 18 ..., irrational number).
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Give two examples to illustrate.
1.0.999999 ...= 1?
The following passage is not for proof, but for understanding-reason: the first step of decimal addition is to align the digits, that is, to know which digit to add and which digit to operate. Adding 0.33333 below ... aligning decimal points with decimal points cannot guarantee the above standards, so it is impossible to add infinite decimals. Since there is no addition, there is no multiplication. )
Everyone knows that 1/3 = 0.333333 ... and both sides are multiplied by 3 at the same time to get 1 = 0.999999 ... but it looks awkward because there is a "finite" number on the left and an "infinite" number on the right.
10×0.999999…… — 1×0.999999……=9=9×0.999999……
∴0.999999……= 1
Second, what is an "irrational number"?
As we know, a number like the root number 2 can't be expressed by the ratio of two integers, and every bit of it has to be determined through constant calculation, and it is endless. This endless number greatly violates people's thinking habits.
Combined with some of the above difficulties, people urgently need a way of thinking to define and study this "endless" number, which leads to the idea of the limit of sequence.
Similar roots are still in physics (in fact, from the course of scientific development, philosophy is the real driving force, but physics has played an unparalleled role in promoting), such as the problem of instantaneous speed. We know that speed can be expressed by the ratio of displacement difference to time difference, and if the time difference tends to zero, this ratio is the instantaneous speed at a certain moment, which raises a question: Is it meaningful to find the ratio of time difference to displacement difference, which tends to be infinitely small, that is, 0÷0 (this meaning refers to the meaning of "analysis", because the geometric meaning is quite intuitive, that is, the tangent slope of this point)? This also forces people to develop a rational explanation for this, and the concept of limit comes to the fore.
The definition of limit in the real modern sense is generally believed to be given by Wilstrass, a middle school math teacher at that time, which is meaningful to our middle school teachers today.