/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 Suppose that \(\lim _{n \rightar... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 25

Suppose that \(\lim _{n \rightarrow \infty} a_{n}=1, \lim _{n \rightarrow \infty} b_{n}=-1\), and \(0<-b_{n}

Short Answer

Expert verified
The limit is 0.

Step by step solution

01

Understand the Limit Conditions

We are given that \( \lim _{n \rightarrow \infty} a_{n}=1 \) and \( \lim _{n \rightarrow \infty} b_{n}=-1 \). Additionally, for all \( n \), \( 0<-b_{n}<a_{n} \). This ensures that both sequences \( a_n \) and \( b_n \) have limits and tells us about the inequalities between them for all \( n \).
02

Express the Target Limit

The limit we need to evaluate is \( \lim _{n \rightarrow \infty} \frac{a_{n}+b_{n}}{a_{n}-b_{n}} \). We'll determine this by substituting the limits of \( a_n \) and \( b_n \) as \( n \to \infty \).
03

Substitute Limits of \( a_n \) and \( b_n \)

Substitute \( \lim _{n \rightarrow \infty} a_{n} = 1 \) and \( \lim _{n \rightarrow \infty} b_{n} = -1 \) into the expression, so we get:\[\lim _{n \rightarrow \infty} \frac{(1) + (-1)}{(1) - (-1)} = \frac{1 - 1}{1 + 1} = \frac{0}{2}.\]
04

Evaluate the Simplified Limit Expression

In the last step, the expression \( \frac{0}{2} \) simplifies directly to zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Limits
Sequence limits are a core concept in calculus where we consider the behavior of sequences as their indices approach infinity. This involves determining the value that a sequence approaches, if it tends toward a single value. As given in the exercise, the sequence \(a_n\) has a limit of 1 and \(b_n\) has a limit of -1 as \(n\) approaches infinity. This means:
  • As \(n\) gets infinitely large, the terms in \(a_n\) get closer and closer to 1.
  • Similarly, the terms in \(b_n\) approach -1.
A limit of a sequence clarifies the "destination" value as the sequence progresses to infinity. This foundational understanding helps us analyze more complex expressions involving these sequences.
Inequality Between Sequences
Understanding the inequality between sequences helps to investigate whether the sequences maintain a specific order as they approach their limits. In this problem, the inequality \(0 < -b_n < a_n\) for all \(n\) means two key things:
  • \(-b_n\) is positive, indicating that \(-b_n\) is closer to zero than \(b_n\) itself, given that \(b_n\) is negative.
  • \(a_n\) is always greater than \(-b_n\), hence more significant when both sequences are nearing their respective limits.
Inequality relations are crucial in assessing the range and comparative behavior of sequences. They ensure that calculations accounting for both \(a_n\) and \(b_n\) align with the fundamental mathematical constraints provided by these limits.
Substitution in Limits
Substitution in limits involves replacing sequence terms with their known limits in calculations to simplify the expression. In this problem, we are tasked with evaluating the limit of the expression \( \lim_{n \to \infty} \frac{a_n + b_n}{a_n - b_n} \). Given:
  • \( \lim_{n \to \infty} a_n = 1 \)
  • \( \lim_{n \to \infty} b_n = -1 \)
We substitute these into the expression. Our mathematical journey leads us to:\[\lim_{n \to \infty} \frac{(1) + (-1)}{(1) - (-1)} = \frac{0}{2} = 0\]This straightforward substitution allows us to focus on manipulating the numerical expressions rather than the sequences themselves. Effectively, this provides an elegant method of simplifying and solving complex limit problems efficiently.

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