① Symbolic function
② Integer function
Represents the largest integer not exceeding x, such as [3.2]=3, and its basic inequality.
③ Dirichlet function
Definition 2. Let the domain of y=f(u) be D f, the domain of u=g(x) be D g, and the domain of value be R g. If D f∩R g≦? The function y=f[g(x)] is called the coincidence function of functions y=f(u) and u=g(x), and its domain is {x|x∈Dg, g(x)∈D f}
Definition 3. Let the domain of function y=f(x) be d and the domain of value be Ry. If there is a unique x∈D for any y=Ry, so that y=f(x), it is denoted as x=f-1 (y), which is called the inverse function of y=f(x).
Definition 4. Power function (y=xμ), exponential function (y=ax), logarithmic function (y=logax), trigonometric function (y=sinx, y=cosx, y=tanx, y=cotx), inverse trigonometric function (y=arcsinx, y=arccosx, y =
If for any two points on the interval I X 1
Common odd functions:
If t > exists; 0, for any x, there is always f(x+T)=f(x), then y=f(x) is called a periodic function, and the smallest positive number t that keeps the above formula is called the smallest positive period.
If there is M>0, so that for any x∈X, there is always |f(x)|≤M, then f(x) is called a bounded function on X.
? ε& gt； 0,? N>0, when n> is n, there is always | x n-a | < ε.
? ε& gt； 0,? X>0, when x> is at X, there is always | f (x)-a | < ε.
? ε& gt； 0,? X>0, when x
? ε& gt； 0,? X>0, when |x| > is at X, there is always | f (x)-a | < ε.
? ε& gt； 0,? δ& gt； 0, when 0 < | x-x0 | < At δ, there is always | f (x)-a | < ε.
If n exists: when n> when n, x n ≤y n ≤z n, and
rule
Monotone bounded sequence must have a limit.
Then α(x) is called the high-order infinitesimal of β(x), which is denoted as α(x)=o[β(x)].
Then α(x) is called the low-order infinitesimal of β(x)
Then α(x) is called the same order infinitesimal of β(x)
Then α(x) is called equal-order infinitesimal of β(x), and it is denoted as α(x)~β(x).
In particular, if
Then α(x) is called the k-order infinitesimal of β(x)
In the same limit process, if f(x) is infinite, then 1/f(x) is infinite. On the other hand, if f(x) is infinitesimal and f(x)≠0, then 1/f(x) is infinite.
If limα(x)=0, limβ(x)=∞, and limα(x)β(x)=A, then lim α (x) β (x) = e a.
And lim α1(x)/β (x) = a ≦-1,then α (x)+β (x) ~ α1(x)+β1(x).
If LIMf(x) = LIMF (x) = 0 (∞), and f(x) and g(x) are derivable in the centripetal field of x0, and g'(x)≠0, limf'(x)/g'(x) exists (or is infinite), then
Where r n (x) = o (x-x 0) n 0) n.
Commonly used inequalities:
Define 1, if
Then it is said that y=f(x) is continuous at point x 0.
If f(x) is defined in the centripetal field of x 0, but it is discontinuous at x 0, then x 0 is called the discontinuous point of f(x).