Function, Limit and Continuity
1. function
(1) Understand the concept of function: definition, representation and piecewise function.
(2) Understand and master the simple properties of functions: monotonicity, parity, boundedness and periodicity.
(3) Understanding the inverse function: the definition and image of the inverse function.
(4) Master the four operations and compound operations of functions.
(5) Understand and master the basic elementary functions: power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function.
(6) Understand the concept of elementary function.
Step 2 limit
(1) Understand the concept of sequence limit: sequence, the definition of sequence limit, and analyze the changing trend of function according to the concept of limit. Will find the left and right limits of a function at a point, and understand the necessary and sufficient conditions for the existence of a function at a point limit.
(2) Understand the nature of limit of sequence: uniqueness, boundedness, four operation theorems, squeezing theorem, monotone bounded sequence, limit existence theorem, and master the four operation rules of limit.
(3) Understand the concept of function limit: the definition of function limit at one point, the left-right limit and its relationship with limit, and the limit of function when X tends to infinity (x→∞, x→+∞, x→-∞).
(4) Mastering the theorem of function limit: uniqueness theorem, squeezing theorem and four operation theorems.
(5) Understanding infinitesimal and infinitesimal: the definition of infinitesimal and infinitesimal, the relationship between infinitesimal and infinitesimal, the properties of infinitesimal and infinitesimal, and the comparison of two infinitesimal orders.
(6) Master the method of finding the limit with two important limits.
3. Continuity
(1) Understand the concept of function continuity: the definition of function continuity at one point, left continuity and right continuity, the necessary and sufficient conditions of function continuity at one point, the discontinuous point of function and its classification.
(2) Grasp the continuity of a function at one point: four operations of a continuous function, the continuity of a composite function and the continuity of an inverse function, find the discontinuous point of the function and determine its type.
(3) Grasp the properties of continuous functions on closed intervals: boundedness theorem, maximum theorem, minimum theorem and intermediate value theorem (including zero point theorem), and use the intermediate value theorem to derive some simple propositions.
(4) Understand that the elementary function is continuous within its defined interval, and use continuity to find the limit.
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