Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 1}\left(2 x^{2}+1\right)=3 $$

Step-by-Step Solution

Answer

The limit is proven: \( \lim_{x \to 1} (2x^2 + 1) = 3 \) using \( \varepsilon-\delta \) argument with \( \delta = \min(1, \varepsilon/6) \).

Step 1: Understanding the Problem

We are given the limit statement \( \lim_{x \to 1} (2x^2 + 1) = 3 \). Our goal is to prove this using an \( \varepsilon-\delta \) argument. This means for any arbitrary \( \varepsilon > 0 \), we need to find a \( \delta > 0 \) such that if \( 0 < |x - 1| < \delta \), then \( |2x^2 + 1 - 3| < \varepsilon \).

Step 2: Express the Limit in Epsilon Terms

Simplify the expression \( |2x^2 + 1 - 3| \). This simplifies to \( |2x^2 - 2| = 2|x^2 - 1| \). We must show that \( 2|x^2 - 1| < \varepsilon \).

Step 3: Factor the Expression

Observe that \( x^2 - 1 = (x - 1)(x + 1) \). Thus, we have \( 2|x^2 - 1| = 2|x - 1||x + 1| \). We need this to be less than \( \varepsilon \).

Step 4: Determine Bounds on x+1

Near \( x = 1 \), we can bound \( |x + 1| \). Let's assume \( \delta \, \leq \, 1 \) so that \( |x - 1| < 1 \). This implies \( x \in (0, 2) \), therefore \( |x + 1| \leq 3 \).

Step 5: Control the Expression with Delta

We want \( 2|x - 1||x + 1| < \varepsilon \). Given \( |x + 1| < 3 \), it suffices to have \( 2|x - 1| \cdot 3 < \varepsilon \), or \( 6|x - 1| < \varepsilon \). This implies that \( |x - 1| < \frac{\varepsilon}{6} \).

Step 6: Concluding Delta Choice

We choose \( \delta = \min(1, \frac{\varepsilon}{6}) \). This ensures both conditions are met: \( |x - 1| < 1 \), allowing \( |x+1| < 3 \) and \( |x - 1| < \frac{\varepsilon}{6} \). Therefore, if \( 0 < |x - 1| < \delta \), then \( |2x^2 + 1 - 3| < \varepsilon \), proving the limit.

Key Concepts

Limits

Limits are a fundamental concept in calculus and mathematical analysis. They describe the behavior of a function as its input approaches a particular value. Imagine you are walking towards a door. As you get closer, you might say you are "approaching" the door. Similarly, in calculus, limits help us understand what happens to a function as the input nears a specific point.

The idea of a limit is captured by the notation \( \lim_{{x \to a}} f(x) = L \), which means that as \( x \) gets closer and closer to \( a \), the function \( f(x) \) approaches \( L \). This concept is crucial when dealing with continuous functions and understanding areas where a function might not be directly defined.

Limits provide the groundwork for other important calculus concepts like derivatives and integrals. They allow us to analyze and predict how functions behave, which is essential for both theoretical mathematics and practical applications in physics, engineering, economics, and more.

Calculus

Calculus is a branch of mathematics focused on change. It deals with how things change and how we can measure these changes. There are two main branches of calculus: differential calculus and integral calculus.

• Differential Calculus: This branch is concerned with the concept of a derivative, which measures how a function changes as its input changes. For example, how fast a car is speeding up or slowing down is a problem for differential calculus.
• Integral Calculus: This part deals with finding areas under curves and summing up infinite series. It allows for the calculation of total accumulated quantities, such as the total distance traveled over time.

Calculus is built on the concepts of limits and infinitesimal change, creating a framework to describe continuous change. It provides powerful tools for solving a variety of real-life problems. Without calculus, we wouldn't have the means to model systems and phenomena across the sciences and engineering effectively.

Mathematical Proofs

Mathematical proofs offer a method to rigorously establish the truth of statements in mathematics. Through logical reasoning, proofs verify or refute mathematical propositions. A proof ensures that a statement holds true in all situations defined by its premises.

An \( \varepsilon-\delta \) proof is a specific type of argument commonly used to prove limits in calculus. This technique provides a precise way to define that a limit exists. Here's a quick breakdown:

• \( \varepsilon \) (epsilon): Represents any small positive number, no matter how tiny, that shows how close the function's output must get to the limit.
• \( \delta \) (delta): Corresponds to the maximum distance from the target value \( x \) can be while still keeping the function's output within \( \varepsilon \) of the limit.

The beauty of mathematical proofs lies in their universal acceptance as evidence. They provide certainty in mathematics, allowing concepts to be built on rock-solid foundations. That's why these proofs are crucial for advancing mathematical thought and ensuring the results are logically sound and irrefutable.