Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give the appropriate \(\varepsilon-\delta\) definition of each statement. $$ \lim _{t \rightarrow a} f(t)=M $$

Step-by-Step Solution

Answer

For every \\(\varepsilon > 0\\), there exists a \\(\delta > 0\\) such that \\(0 < |t - a| < \delta\\) implies \\(|f(t) - M| < \varepsilon\\).

Step 1: Introduction to the Concept

In the context of limits, the \(\varepsilon-\delta\) definition is a precise way of saying that a function \(f(t)\) approaches the limit \(M\) as \(t\) approaches \(a\). Our goal is to express this limit in terms of two positive numbers: \(\varepsilon\) and \(\delta\).

Step 2: Identify the Target Condition

For \(\lim _{t \rightarrow a} f(t) = M\), we need to ensure that the values of \(f(t)\) are arbitrarily close to \(M\) whenever \(t\) is sufficiently close to \(a\), but not equal to \(a\).

Step 3: Define the \\(\varepsilon\\)

Choose \(\varepsilon > 0\), which represents the allowable deviation from \(M\) for the function \(f(t)\). Our task is to find a corresponding \(\delta\).

Step 4: Determine the \\(\delta\\) Condition

Find \(\delta > 0\) such that whenever \(0 < |t - a| < \delta\), it ensures \(|f(t) - M| < \varepsilon\). This \(\delta\) keeps \(f(t)\) close to \(M\) whenever \(t\) is near \(a\).

Step 5: Compose the Precise Definition

By combining the conditions, the \(\varepsilon-\delta\) definition for the given limit is: For every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |t - a| < \delta\), then \(|f(t) - M| < \varepsilon\).

Key Concepts

Limits in Calculus

Limits are a fundamental concept in calculus that help us understand the behavior of functions as they approach a particular point. By studying limits, we can see what value a function is getting closer to as the input gets closer to a certain number.
In simpler terms, think of a limit as the destination a function aims to reach, even though it might never actually get there. Imagine you're walking towards a point on a map. A limit helps tell you exactly which point you're trying to reach, no matter how tiny the steps you must take.
Understanding limits is crucial because many calculus concepts, like derivatives and integrals, are built on the notion of limits. Limits provide the foundation that supports the more advanced topics in calculus.

Limit of a Function

When we talk about the limit of a function, we mean the value the function is approaching as the input approaches a specific point. This is denoted as \( \lim_{t \to a} f(t) = M \), where \( f(t) \) is a function, \( a \) is the point the input \( t \) is getting closer to, and \( M \) is the value \( f(t) \) is approaching.
Let's break this down:

• \( t \to a \) indicates that \( t \) is getting very close to \( a \), but it doesn't have to reach \( a \) exactly.
• \( f(t) = M \) suggests that as \( t \) nears \( a \), \( f(t) \) approaches the value of \( M \).

This concept helps us understand how functions behave in scenarios where direct computation may be tricky or impossible. By knowing a limit, we can anticipate a function's behavior near crucial points without having to plug in every value.

Precise Definition of Limits

The precise definition of limits, also known as the \( \varepsilon-\delta \) definition, is a rigorous way to mathematically define when a function is approaching a limit. The goal is to capture the idea that a function's output can be made arbitrarily close to a limit by restricting how close the input is to a certain point.
Here's how it works:

• The symbol \( \varepsilon \) (epsilon) represents how close \( f(t) \) needs to be to \( M \). You can think of it as a margin of error that you are allowing for \( f(t) \).
• \( \delta \) (delta) indicates how close \( t \) needs to be to \( a \). It's the range within which \( t \) must fall to keep \( f(t) \) within the \( \varepsilon \) margin.
• We say that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |t - a| < \delta \), it ensures \( |f(t) - M| < \varepsilon \).

This definition is essential for precisely ensuring the behavior of functions near particular points and is foundational in the study of calculus. It provides a structured way to prove the limits of functions formally and accurately.