definition of limit

properties of limits

The ultimate algorithm includes four arithmetic operations.

The operation rules of the limit need to follow the properties of real numbers, such as finiteness, additivity, multiplication, etc.

The operation rules of the limit involve the basic concepts of calculus, such as derivatives, continuity, original functions, etc.

The algorithm of limit has important application value in solving differential equations, integral equations, etc.

The extreme algorithm needs to be analyzed and applied in conjunction with specific problems and cannot be applied mechanically.

Continuous function: Within a certain interval, the function value is infinitely close to a constant as the independent variable increases or decreases.

Left limit and right limit: The left limit of a function at a certain point means that when the independent variable approaches the left side of the point, the function value approaches the limit value of the point; the right limit means that when the independent variable approaches the point When on the right side of , the function value approaches the limit value of that point.

Definition of continuity: If the left limit and right limit of a function in a certain interval exist and are equal, then the function is continuous in that interval.

Continuity theorem: If a function is continuous at every point in an interval, then the function is also continuous in that interval.

Infinite small quantity and infinite large quantity: Infinite small quantity refers to the quantity whose limit value is 0 when the independent variable approaches 0; infinite quantity refers to the quantity whose limit value does not exist when the independent variable approaches positive infinity or negative infinity.

Properties of continuity: The derivative of a continuous function at a certain point is equal to the slope of the tangent line at that point; if f(x) is continuous within a certain interval, then f'(x) is also continuous within the interval.

The addition rule of continuous functions: If f(x) and g(x) are both continuous functions, then f(x) g(x) is continuous on the interval [a, b].

Multiplication rule for continuous functions: If f(x) and g(x) are both continuous functions, then f(x)×g(x) is continuous on the interval [a, b].

Division rule for continuous functions: If f(x) and g(x) are both continuous functions, and g(x) is not equal to 0, then f(x)÷g(x) is continuous on the interval [a, b].

The composite function rule of continuous functions: If f(x) is a continuous function and g(x) is also a continuous function, then h(x)=f(g(x)) is continuous on the interval [a,b].

The relationship between limit and continuity: If f(x) tends to a certain value C on the closed interval [a, b], then f(x) must be continuous on this interval.

Basic concepts of limit and continuity: Limit is a concept in mathematics that describes the changing trend of a function near a certain point, and continuity is one of the necessary conditions to ensure the existence of a limit.

Definition of the derivative of a continuous function: Assume f(x) is a continuous function, then the derivative at point a exists and is unique, denoted as f'(a), which represents the rate of change of the function at point a.