Problem 74 - Calculus for Biology and Medicine

Question

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 \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n+1}=0 $$

Step-by-Step Solution

Answer

For every \( \epsilon > 0 \), choose \( N = \lceil \frac{1}{\epsilon} - 1 \rceil \) to satisfy the condition \( \left| \frac{1}{n+1} \right| < \epsilon \) whenever \( n > N \).

Step 1: Understand the Limit Definition

The goal is to show that for every \( \epsilon > 0 \), there exists a sufficiently large integer \( N \) such that for all \( n > N \), the inequality \( \left| \frac{1}{n+1} - 0 \right| < \epsilon \) holds. This will confirm that \( \lim_{n \to \infty} \frac{1}{n+1} = 0 \).

Step 2: Analyze the Expression

The expression \( \left| \frac{1}{n+1} - 0 \right| \) simplifies to \( \frac{1}{n+1} \). We need to find \( N \) such that \( \frac{1}{n+1} < \epsilon \) for every \( n > N \).

Step 3: Solve the Inequality

We need \( \frac{1}{n+1} < \epsilon \), which means that \( n+1 > \frac{1}{\epsilon} \). Therefore, we can rearrange this inequality to find \( n > \frac{1}{\epsilon} - 1 \).

Step 4: Determine N

Based on the inequality \( n > \frac{1}{\epsilon} - 1 \), we determine \( N \) by setting \( N = \left\lceil \frac{1}{\epsilon} - 1 \right\rceil \), where \( \lceil \cdot \rceil \) represents the ceiling function. This ensures that \( n > N \) satisfies the desired inequality.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define the limit of a function or sequence. It provides a precise method to understand how functions behave as they approach a particular point, often infinity. This concept is crucial in calculus and mathematical analysis.
To grasp it, consider a function \( f(x) \) that approaches a limit \( L \) at \( x = a \). The definition states:

• For every \( \epsilon > 0 \) (no matter how small), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \epsilon \).
• The \( \epsilon \) represents the arbitrary closeness to the limit \( L \).
• The \( \delta \) corresponds to how close \( x \) needs to be to \( a \) to achieve this closeness.

This framework helps in proving the continuity of functions and ensures their behavior is as expected near specific points. In simpler terms, it means we can make \( f(x) \) as close to \( L \) as we want, provided \( x \) is sufficiently close to \( a \). This concept is particularly important when considering limits at infinity, as it assists in determining asymptotic behavior for functions and sequences.

Sequence Convergence

Sequence convergence is about how sequences behave as they head towards infinity. Specifically, a sequence \( \{a_n\} \) converges to a limit \( A \) if the terms get arbitrarily close to \( A \) as \( n \) increases.
Formally, the sequence \( \{a_n\} \) converges to \( A \) if for every \( \epsilon > 0 \), we can find a natural number \( N \) such that \( |a_n - A| < \epsilon \) for all \( n > N \).
This definition means:

• The terms of the sequence get closer and closer to \( A \) as the sequence progresses.
• Beyond a certain point, all terms are within \( \epsilon \) distance of \( A \), no matter how tiny the \( \epsilon \) chosen.

An example seen in the original exercise is \( \lim_{n \to \infty} \frac{1}{n+1} = 0 \). Here, the sequence \( \frac{1}{n+1} \) converges to 0 because as \( n \) gets larger, \( \frac{1}{n+1} \) becomes tinier, essentially approaching zero.

Limit of a Function

The limit of a function is fundamental to understanding calculus. It describes the behavior of a function as its input approaches a particular value.
The limit \( \lim_{x \to a} f(x) = L \) signifies that as \( x \) gets closer to \( a \), \( f(x) \) approaches the value \( L \).
This concept is crucial for tackling questions on continuity, derivatives, and integrals. The limit can be approached from either side of \( a \):

• Left-sided limit: \( \lim_{x \to a^-} f(x) \).
• Right-sided limit: \( \lim_{x \to a^+} f(x) \).

A function's limit at infinity, \( \lim_{x \to \infty} f(x) \), tells us about the function's asymptotic behavior. For the given exercise, the limit \( \lim_{n \to \infty} \frac{1}{n+1} = 0 \) is established using the epsilon-delta definition, showcasing convergence to zero and illustrating how functions simplify as inputs grow very large. Understanding how a function behaves as it approaches a limit helps predict values and solve complex calculus problems comprehensively.