Problem 70 - Calculus for Biology and Medicine

Question

Formal Definition of Limits: Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{n}{n+1}=1 $$

Step-by-Step Solution

Answer

Choosing \( N = \left\lfloor \frac{1}{\epsilon} \right\rfloor \) satisfies \( \left| \frac{n}{n+1} - 1 \right| < \epsilon \) for all \( n > N \), proving the limit is \( 1 \).

Step 1: Understanding the Limit Definition

The limit definition states that for any real number \( \epsilon > 0 \), there exists an integer \( N \) such that for all \( n > N \), the inequality \( \left| a_n - a \right| < \epsilon \) holds. Here, \( a_n = \frac{n}{n+1} \) and \( a = 1 \). We need to show \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \) by this definition.

Step 2: Finding the Expression to be Less than Epsilon

We have \( \left| \frac{n}{n+1} - 1 \right| < \epsilon \). Simplifying this gives us \( \left| \frac{n - (n+1)}{n+1} \right| = \left| \frac{-1}{n+1} \right| = \frac{1}{n+1} < \epsilon \). Now we need to find \( N \) such that \( \frac{1}{n+1} < \epsilon \).

Step 3: Solving the Inequality

Start with the inequality \( \frac{1}{n+1} < \epsilon \). Rearranging gives \( n+1 > \frac{1}{\epsilon} \). Solving for \( n \), we get \( n > \frac{1}{\epsilon} - 1 \).

Step 4: Choosing N to Satisfy the Condition

Since we need \( n > N \), choose \( N = \left\lfloor \frac{1}{\epsilon} \right\rfloor \). For all \( n > N \), it will ensure that \( \frac{1}{n+1} < \epsilon \) is satisfied.

Step 5: Conclusion with the Formal Definition

We have found that for any \( \epsilon > 0 \), by choosing \( N = \left\lfloor \frac{1}{\epsilon} \right\rfloor \), \( \left| \frac{n}{n+1} - 1 \right| < \epsilon \) whenever \( n > N \). Therefore, \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \).

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition of a limit is a fundamental concept in calculus that provides a rigorous way to define what it means for a sequence to approach a certain value. In layman's terms, when we say that a sequence \( a_n \) converges to a limit \( a \), we mean that no matter how close you want \( a_n \) to be to \( a \) (determined by the positive number \( \epsilon \)), there is always a point in the sequence (determined by \( N \)) where every subsequent term \( a_n \) is within \( \epsilon \) of \( a \).

• Choose any small \( \epsilon > 0 \).
• Find an integer \( N \) such that whenever \( n > N \), the difference \( \left| a_n - a \right| < \epsilon \).
• This ensures that the values of \( a_n \) get closer and closer to \( a \) as \( n \) increases.

In the given exercise, using the epsilon-delta definition, we show that \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \) by finding an appropriate \( N \) for any \( \epsilon > 0 \). This rigorous approach not only proves convergence but also gives bounds on how fast the sequence approaches the limit.

Convergence

Convergence is a key concept when dealing with limits. It indicates that as you move through a sequence, the terms increasingly get close to a specific number, known as the limit. Think of convergence as a journey, where as you move along, you get nearer to a destination. The distance to this destination decreases as you progress through more terms in a sequence.
Here’s what convergence means in practical terms:

• A sequence \( a_n \) converges to \( a \) if for every small number \( \epsilon > 0 \), we can find a point in the sequence (i.e., \( N \)) beyond which all terms are arbitrarily close to \( a \).
• In the example \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \), as \( n \) becomes larger and larger, \( \frac{n}{n+1} \) gets closer to 1.
• Visually, if plotted on a graph, the values of the sequence would hover closer to the horizontal line at \( y = 1 \) as \( n \to \infty \).

Understanding convergence helps in determining the behavior of sequences and functions as they extend into infinity, and plays a crucial role in calculus and mathematical analysis.

Infinite Limits

Infinite limits describe the behavior of functions or sequences as they extend towards infinity. These limits help us understand what value, if any, a function approaches as the input grows very large. It's as if you’re traveling towards a horizon that represents a particular number.
In the context of sequences, infinite limits highlight two key situations:

• When \( n \to \infty \), and the sequence terms approach a specific finite value (like \( \lim_{n \to \infty} \frac{n}{n+1} = 1 \)).
• When the sequence terms themselves grow larger without bound (like \( \lim_{n \to \infty} n = \infty \)).

The concept of infinite limits ties back to both the epsilon-delta definition and convergence. For a sequence like \( \frac{n}{n+1} \) that approaches 1 as \( n \to \infty \), understanding infinite limits helps determine how tightly the values hug the "horizon" at \( y = 1 \).
Through infinite limits, we gain the ability to predict and articulate the eventual behavior of sequences or functions, which is essential in fields like calculus, engineering, and physics.