Lecture 9: Limits — Concepts, Properties, and Existence Criteria
Graduate Entrance Examination Mathematics study notes: Lecture 9: Limits — Concepts, Properties, and Existence Criteria. Original formulas, diagrams, and examples are retained.
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9.1 The Concept of Limits
Example Problem Introduction
- For any given $\varepsilon\in(0,1)$, there always exists a positive number $N$. When $n>N$, there is always $|x_n-\boldsymbol{a}|\leq 2\boldsymbol{\varepsilon}$ ${x_n}$ the field of convergence of the sequence. What condition is this?
- It is a sufficient and necessary condition;
- 1. When $x_n$ converges to a, the difference between the two should be arbitrarily small, tending toward zero;
- 2. In $|x_n-\boldsymbol{a}|\leq 2\boldsymbol{\varepsilon}$, the $\varepsilon$ citation can take any value on $(0,1)$, including always tending toward 0, so it can be arbitrarily small, even multiplied by 100 or 1000 is arbitrarily small ->, so it can be deduced that $x_n$ converges to a;
- 3. Similarly, when $x_n$ converges to a, the value of $|x_n-\boldsymbol{a}|$ is infinitesimal, while $2{\varepsilon}$ tends toward infinitesimal;