First, prove it with pinch theorem.

Secondly, the application of monotonicity is proved by definition.

Third, start with the definition of limit to prove it.

4. Proof of the necessary and sufficient conditions for the existence of application limit.

Function, a mathematical term. Its definition is usually divided into traditional definition and modern definition. The essence of these two functional definitions is the same, but the starting point of narrative concept is different. The traditional definition is from the perspective of movement change, and the modern definition is from the perspective of set and mapping.

The modern definition of a function is to give a number set A, assume that the element in it is X, apply the corresponding rule F to the element X in A, and record it as fx to get another number set B, assume that the element in B is Y, and the equivalent relationship between Y and X can be expressed as y=f(x). The concept of a function includes three elements: the domain A, the domain B and the corresponding rule F, among which the core is the corresponding rule F, which is the essential feature of the function relationship.

Origin of function

The word "function" used in China's math book is translated. It was Li, an algebra expert in China's Qing Dynasty, who translated "function" into "function" when he translated the book Algebra 1859.

In ancient China, the word "Xin" and the word "Han" were universal and both had the meaning of "Han". Li's definition is: "every formula contains heaven, which is a function of heaven." In ancient China, four different unknowns or variables were represented by four words: heaven, earth, people and things. The meaning of this definition is: "Whenever a formula contains the variable X, the formula is called a function of X."

So "function" means that the formula contains variables. The exact definition of an equation refers to an equation containing unknowns. However, in the early mathematical monograph "Nine Chapters Arithmetic" in China, the word "equation" means simultaneous linear equations with many unknowns, which are called linear equations.