A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…

Verified step by step guidance

1. Identify the sequence function: f(n) = \(\frac{n-1}{n}\).
2. Express the limit of the sequence as n approaches infinity: \(\lim\)_{n \(\to\) \(\infty\)} \(\frac{n-1}{n}\).
3. Simplify the expression: \(\frac{n-1}{n}\) = \(\frac{n}{n}\) - \(\frac{1}{n}\).
4. Recognize that \(\frac{n}{n}\) simplifies to 1 and \(\frac{1}{n}\) approaches 0 as n approaches infinity.
5. Apply the limit laws: \(\lim\)_{n \(\to\) \(\infty\)} \(\left\)(1 - \(\frac{1}{n}\)\(\right\)) = 1 - 0.

Key Concepts

Sequences

A sequence is an ordered list of numbers that can be defined by a specific function. Each term in the sequence corresponds to a natural number, and sequences can be finite or infinite. Understanding sequences is crucial for analyzing their behavior, especially as the index approaches infinity.

Limits

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. It is denoted as lim n→∞ f(n). If the terms converge to a specific value, that value is the limit; if they do not settle at any value, the limit does not exist. This concept is fundamental in calculus for understanding the behavior of functions and sequences.

Limit Laws

Limit laws are a set of rules that simplify the process of finding limits. They include properties such as the sum, product, and quotient of limits, which can be applied to sequences. These laws help in determining the limit of complex sequences by breaking them down into simpler components, making it easier to analyze their behavior as n approaches infinity.