1. Infinity: Infinity means that the value of a series or function is infinitely close to a certain number or approaches infinity. Common infinity includes positive infinity (+∞) and negative infinity (-∞). Infinite quantity plays an important role in limit operation. For example, when the terms of a series gradually approach infinity, we can describe the behavior of this series with limits.
2. Bounded quantity: Bounded quantity is the value of an exponential sequence or function within a certain range. In other words, the bounded quantity does not increase or decrease indefinitely, but has an upper limit and/or a lower limit. Bounded quantities are very important in mathematical analysis because they guarantee the establishment of some properties and theorems. For example, for a bounded function, there must be a maximum and a minimum.
3. Infinite classification: According to the infinite trend, it can be divided into the following categories:
-Infinite oscillation: the value of an exponential sequence or function swings back and forth in a certain interval, but does not tend to infinity or infinitesimal.
-Infinite convergence: the value of exponential sequence or function tends to a certain number or to infinity/infinitesimal.
-Infinite divergence: it means that the value of a series or function tends to infinity/infinitesimal, but not to a certain number.
4. Attributes of bounded quantity: Bounded quantity has the following attributes:
Bounded quantities must be bounded, that is, there is an upper bound and/or a lower bound.
-If the function is bounded, so is its inverse function.
-If a function is bounded and its derivative is bounded, then the function must be continuous.
5. Relationship between infinite quantity and bounded quantity: There is a certain relationship between infinite quantity and bounded quantity. For example, if a series is bounded, its limit may be a finite value, infinite or infinitesimal. Similarly, if a function is bounded, its limit may be a finite value, infinite or infinitesimal.
In a word, infinite quantity and bounded quantity are important basic concepts in mathematics, which are widely used in mathematical analysis and other fields. Understanding these concepts and their properties is very important for understanding and solving various mathematical problems.