Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 0} x^{4}=0 $$

Step-by-Step Solution

Answer

For every \(\varepsilon > 0\), choose \(\delta = \varepsilon^{1/4}\). Then \(|x^4| < \varepsilon\) whenever \(|x| < \delta\).

Step 1: Understand the Definition

To give an \(\varepsilon-\delta\) proof for \(\lim _{x \rightarrow 0} x^{4}=0\), we need to show that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(|x - 0| < \delta\), then \(|x^4 - 0| < \varepsilon\). This sets up our goal for the proof.

Step 2: Set Up the Inequality

We need to show \(|x^4| < \varepsilon\). Since \(|x^4| = |x|^4\), our task is to find a \(\delta\) in terms of \(\varepsilon\) such that \(|x|^4 < \varepsilon\) whenever \(|x| < \delta\).

Step 3: Determine \(\delta\)

To ensure \(|x|^4 < \varepsilon\), we choose \(\delta = \varepsilon^{1/4}\). This choice implies that \(|x| < \delta\) leads to \(|x|^4 = |x|^4 < \varepsilon^{1/4}^4 = \varepsilon\).

Step 4: Conclude the Proof

Given \(|x| < \varepsilon^{1/4}\), it follows that \(|x|^4 < \varepsilon\). This satisfies the conditions for an \(\varepsilon-\delta\) proof. Therefore, \(\lim _{x \rightarrow 0} x^{4}=0\) is proven.

Key Concepts

Limit of a Function

Understanding the limit of a function is fundamental in calculus and mathematical analysis. When we say that the limit of a function \( f(x) \) as \( x \) approaches a certain value is \( L \), we mean that as \( x \) gets closer and closer to this value, \( f(x) \) gets closer and closer to \( L \). This concept can be expressed formally using epsilon-delta (\(\varepsilon-\delta\)) definitions.
- Here, \(\varepsilon\) (epsilon) represents any small positive number we choose.- The goal is to keep \( f(x) \) within an epsilon distance from \( L \) whenever \( x \) is sufficiently close to the specified value, but not exactly equal to it.- The corresponding \(\delta\) ensures that \( x \) values are close enough to cause \( f(x) \) to be near \( L \).For the limit \( \lim_{x \to 0} x^4 = 0 \), our task is to show this closeness using epsilon-delta conditions.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with limits, continuity, derivatives, and integrals. It is a rigorous field that provides the tools needed to understand the behavior of functions in a deeply precise manner.

• Central to mathematical analysis is the concept of limits and how we approach them.
• It often requires presenting arguments with logical precision and structure.
• The epsilon-delta proof directly stems from this precision, ensuring all conditions of the limit are met using carefully constructed logical steps.

Let's consider a function \( f: x \mapsto x^4 \) and see how mathematical analysis verifies its limit as \( x \) approaches 0. The ability to define such a limit using precise conditions enhances the understanding of function continuously and its behavior in specific intervals.

Continuity

Continuity is a crucial concept in calculus, closely tied to the behavior of functions. A function is continuous at a point if small changes in input result in small changes in the output. This behavior means that the graph of the function has no holes, jumps, or breaks at that point.
- If you were to trace the graph of a continuous function, you could do so without lifting your pencil off the paper.- For a function to be continuous at a particular point, the limit of the function as it approaches that point must equal the function's value at that point.For example, the function \( x^4 \) is continuous at \( x = 0 \), as proven by the epsilon-delta method. Showing that its limit equals its value at that point validates its continuity.

Calculus Concepts

Calculus is a gateway to studying change and motion, through two main branches: differential and integral calculus. Differential calculus focuses on concepts like the derivative, while integral calculus is about areas under curves and accumulating quantities.

• The epsilon-delta approach is a fundamental aspect of dealing with limits within differential calculus.
• Understanding this concept aids in mastering derivatives and integrals, which are built upon limits.
• Garnering a good grounding in these basics prepares you for advanced calculus topics like series, multivariable functions, and vector calculus.

By practicing proofs like \( \lim_{x \to 0} x^4 = 0 \), students get to see the application of these calculus concepts in verifying functional behaviors accurately and rigorously.