Question

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

Short answer

The limit is \(+\infty\).

Step-by-step solution

Identify the Problem

We need to find the limit \( \lim _{n \rightarrow \infty} \frac{a_{n}-b_{n}}{a_{n}+b_{n}} \). We are given that \( \lim_{n \rightarrow \infty} a_n = 1 \) and \( \lim_{n \rightarrow \infty} b_n = -1 \), with the condition \( 0 < -b_n < a_n \) for all \( n \).

Examine the Numerator

The numerator of the given fraction is \( a_n - b_n \). Using the limits, as \( n \rightarrow \infty \), \( a_n \rightarrow 1 \) and \( b_n \rightarrow -1 \). Therefore, \( a_n - b_n \rightarrow 1 - (-1) = 2 \).

Examine the Denominator

The denominator is \( a_n + b_n \). As \( n \rightarrow \infty \), \( a_n \rightarrow 1 \) and \( b_n \rightarrow -1 \), thus \( a_n + b_n \rightarrow 1 + (-1) = 0 \).

Analyze the Limit Using the Form "+ Infinity/0"

Since the calculation leads to a form \( \frac{2}{0} \), this implies the fraction approaches \(+\infty\) or \(-\infty\), depending on how \( a_n + b_n \) approaches 0. However, further analysis shows that since \( a_n > -b_n \), \( a_n + b_n \) is always positive, approaching 0 from the positive side.

Determine the Limit

Given that the numerator approaches a positive constant and the denominator approaches 0 from the positive side, the limit approaches \(+\infty\). Thus, \( \lim _{n \rightarrow \infty} \frac{a_{n}-b_{n}}{a_{n}+b_{n}} = +\infty \).

Key concepts

Sequences

In calculus, sequences are ordered lists of numbers, much like a story unfolds chapter by chapter. Each term in a sequence is represented by a position number, often denoted as \(n\). As \(n\) increases, the terms of the sequence can approach a certain value, and this concept is fundamental in calculus for understanding limits. Sequences can be finite or infinite, and they are the building blocks for more complex mathematical constructs like series.

To better understand sequences, think of them as a bridge between individual numbers and the continuous nature of functions. For example, the sequence given in the exercise, \(a_n\), approaches the number 1 as \(n\) becomes very large. This is expressed as \(\lim_{n \rightarrow \infty} a_n = 1\). Similarly, \(b_n\) approaches -1.

• Understanding how sequences behave helps us predict the outcome when they interact with each other.
• This interaction forms the basis for evaluating limits, as seen in the exercise where \(a_n - b_n\) and \(a_n + b_n\) converge to specific values.

Infinity

Infinity is more than just a big number; it's a concept that describes something without any bounds. In mathematics, we often use the symbol \( \infty \) to represent this idea, especially when discussing limits and sequences. This concept allows mathematicians to explore the behavior of sequences or functions as they grow beyond any finite bound.

In the exercise, as \(n\) approaches infinity, both sequences \(a_n\) and \(b_n\) reach their respective limits at different values. However, when analyzing the limit of \(\frac{a_n - b_n}{a_n + b_n} \), we encounter a situation where the denominator approaches zero, and the numerator approaches a finite positive constant, leading our limit towards infinity.

• This contrast forms an essential part of calculus, showing how infinity can emerge from the interaction of sequences.
• Infinity helps us understand behaviors at extremes, offering a way to handle mathematical operations that seem impossible at first glance.

Numerical Analysis

Numerical analysis is the study of algorithms that use numerical approximation to solve mathematical problems. This branch of mathematics is crucial for dealing with real-world problems where exact solutions are hard or impossible to determine.

In the given exercise, numerical analysis might come into play when calculating approaching values for limits involving sequences. Here, the limit of the expression \(\frac{a_n - b_n}{a_n + b_n}\) as \(n \to \infty\) illustrates clear deductions through step-by-step calculations.

• Numerical analysis helps break down complex mathematical problems into computational steps, enabling predictions about sequences as they extend towards infinity.
• It teaches how to handle division by zero situations by assessing how the numerator and denominator behave independently as \(n\) grows.

This kind of analysis is vital for predicting trends and outcomes with precision and is widely used across science, engineering, and finance.