Lecture 7: Criteria for the Existence of a Limit

7.1 Basic Concepts of the Criterion of Limit Existence

7.1.1 Criteria for the existence of sequence limits

#####Theorem: Squeezingcriterionforsequencelimits

Description: If exists $N, $ when $n>N$, $x_n\leq y_n\leq z_n$, $\lim_{n\to\infty}x_n=\lim_{n\to\infty}z_n=a, $ then $\lim_{n\to\infty}y_n=a$

Usage Scenario: n items and

• Example: $\lim_{n\to\infty}\left[\frac n{n^2+1}+\frac n{n^2+2}+\cdots+\frac n{n^2+n}\right]$
• Magnify: $\frac n{n^2+n}$
• Shrink: $\frac n{n^2+1}$
• Usage: $\frac{n^2}{n^2+n}\leq\left[\frac n{n^2+1}+\frac n{n^2+2}+\cdots+\frac n{n^2+n}\right]\leq\frac{n^2}{n^2+1}$

Usage: floor function

• Example: $\lim_{x\to0^+}x[\frac1x]$
• Size: $\frac1x-1<[\frac1x]\leq\frac1x$

#####Theorem: Themonotonicboundedcriterionforsequences

description: monotonically bounded sequences must have limits;

1. Sequences with monotonic increasing or upper bounds must have limits;

2. Sequences with monotone subtraction or lower bounds must have limits;