Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow-21}(3 x-1)=-64 $$

Step-by-Step Solution

Answer

Use the \(\varepsilon-\delta\) definition to show for every \(\varepsilon > 0\), there is a \(\delta > 0\) such that \(0 < |x + 21| < \delta\), ensures \(|3x - 1 + 64| < \varepsilon\).

Step 1: Understand the Limit Statement

The limit statement \( \lim_{x \to -21}(3x - 1) = -64 \) implies that as \( x \) approaches \( -21 \), the function \( 3x - 1 \) approaches \( -64 \). This can be proven using the \( \varepsilon-\delta \) definition of a limit.

Step 2: Recall the \(\varepsilon-\delta\) Definition

The \( \varepsilon-\delta \) definition of a limit says that \( \lim_{x \to a} f(x) = L \) if for every \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |f(x) - L| < \varepsilon \).

Key Concepts

Understanding Calculus Limits

Calculus limits help us understand how functions behave as inputs approach certain values. In the given problem, the function we have is a linear function, specifically, \( f(x) = 3x - 1 \). The limit tells us what value the function approaches when \( x \) gets very close to \( -21 \). Here, the limit specifies that this value is \( -64 \). This concept is fundamental in calculus because it allows us to analyze the behavior of functions at points where they may not be directly defined.

To comprehend limits, think about zooming in on the graph close to \( x = -21 \). You would notice that \( f(x) \) draws closer to \( -64 \) the nearer \( x \) gets to \( -21 \). The strength of calculus comes from its ability to formalize this intuitive concept with precision.

Delta-Epsilon Proof Methodology

Delta-Epsilon proofs provide a rigorous way to establish limits. The aim is to prove that given any small \( \varepsilon > 0 \), determining how close \( x \) needs to be to \( -21 \) (using \( \delta \)) ensures \( |3x - 1 + 64| < \varepsilon \). This formalizes the intuitive notion of approaching a value.

The steps in a Delta-Epsilon proof work like this:

• Choose an arbitrary small number for \( \varepsilon \).
• Find \( \delta \) so that whenever \( |x + 21| < \delta \), you have \( |3x + 63| < \varepsilon \).
• Work algebraically to express \( \delta \) in terms of \( \varepsilon \).

The essence of this proof involves showing that for a prescribed distance in function values (\( \varepsilon \)), there exists a neighborhood around \( x = -21 \) (defined by \( \delta \)) where the condition holds. It ensures the defined closeness vertically comes from a controlled horizontal proximity.

Exploring Limit Proof Techniques

Limit proof techniques, such as the Delta-Epsilon method, are fundamental in verifying the truth of limit statements. They help us understand why a function approaches a particular value by dissecting the function’s behavior.

It's crucial to develop a strategy that includes:

• Recognizing patterns and using algebraic manipulations to your advantage.
• Choosing values methodically for \( \varepsilon \) to simplify calculations.
• Transforming expressions logically and using inequalities effectively.

For instance, once you select an appropriate \( \varepsilon \), the subsequent steps involve ensuring that you can choose a \( \delta \) that implies \( |3x - 1 - (-64)| < \varepsilon \). This proves, with certainty, the function \( 3x - 1 \) nears \( -64 \) as \( x \) approaches \( -21 \). Mastery of these proof techniques solidifies understanding by offering a framework to think critically and tackle more complex limits.