What is a function?
Let x and y be two variables, D is a given set of non-empty numbers, if according to a corresponding law f, for each number x ∈ D, the variable y has a unique value corresponding to it, then Call this corresponding law f as a function defined on D. The number set D is called the domain of this function, x is called the independent variable, and y is called the dependent variable.
Some specific functions: power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function, constant function, absolute value function, sign function sgn, rounding function, piecewise function.
Four properties of functions: boundedness, monotonicity, parity, periodicity.
What is the limit of the function?
The limit of the function when the independent variable tends to a finite value:
suppose that the function f(x) is defined in a decentered neighborhood of the point x0, if there is a constant A, so that for any given positive number ε (no matter how small it is ), there is always a positive number δ, as long as the point x fits the inequality 0<|x-x0|<δ, and the corresponding function value f(x) satisfies the inequality |f(x)-A|<ε, then the constant A is called f(x) is the limit when x→x0. If such a constant does not exist, then f(x) has no limit when x→x0 is called.
The limit
of the function when the independent variable tends to infinity: Let the function f(x) be defined when |x|>M (M is a positive number). If there is a constant A, so that for any given positive number ε (no matter how small it is), there is always X, as long as the independent variable x fits the inequality |x|>X, the corresponding function value f(x) satisfies the inequality | f(x)-A|<ε, then the constant A is called the limit of the function f(x) when x→∞. If such a constant does not exist, then f(x) has no limit when x→∞.
Local sign preservation of the limit:
if 
, and A≠0, then there is a positive number δ (or positive number X), when 0<|x-x0|<δ(or |x|>X), f (x) Constant is not zero and has the same sign as A. This property shows that: in a local variation range of the independent variable, the function value f(x) and the limit value A keep the same sign.