Problem 40 Prove the statement using the \(... [FREE SOLUTION]

Chapter 1: Problem 40

Prove the statement using the $\varepsilon, \delta$ definition of a limit. $$\lim _{x \rightarrow 0} x^{3}=0$$

Short Answer

Expert verified

Choose $\delta = \varepsilon^{1/3}$, then $|x^3| < \varepsilon$ for $|x| < \delta$.

Step by step solution

Understanding the Statement

We need to prove that as $x$ approaches 0, $x^3$ approaches 0 as well. In mathematical words, given any $\varepsilon > 0$, there should be a $\delta > 0$ such that if $|x| < \delta$, then $|x^3| < \varepsilon$.

Express |x^3| < 蔚

According to the $\varepsilon, \delta$ definition, we want $|x^3| < \varepsilon$ whenever $|x| < \delta$. Notice that $|x^3| = |x|^3$. We need to find a relation between $|x|$ and $\varepsilon$.

Choose 未 in terms of 蔚

To achieve $|x|^3 < \varepsilon$, take the cube root of both sides: $|x| < \varepsilon^{1/3}$. Hence, we can choose $\delta = \varepsilon^{1/3}$.

Verification

Verify that when $|x| < \delta$, the condition $|x^3| < \varepsilon$ is satisfied. By substitution, $|x| < \delta = \varepsilon^{1/3}$ implies $|x|^3 < (\varepsilon^{1/3})^3 = \varepsilon$. This shows the solution works as intended.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Epsilon-Delta Definition

In calculus, the epsilon-delta definition is critical for rigorously defining limits. It provides a formal approach to what it means for a function to approach a certain value as the input approaches some point. In simple terms:

For every small distance (epsilon, $\varepsilon$), you want the function's output to be within that distance from the target value.
You achieve this by picking a suitable boundary (delta, $\delta$) for the input.

The objective is to find a delta such that whenever the distance between the input, $x$, and the approaching value (say 0) is less than delta, the distance between the function's output and the limit is less than epsilon.
This creates a framework within which function behavior is tightly controlled and predictable near the limit point.

Limit as x Approaches 0

Understanding the process of finding a limit as $x$ approaches 0 is fundamental. It means we observe how a function behaves when the input values get closer and closer to zero.
For example, for the function $x^3$, we are interested in proving that as $x$ approaches 0, $x^3$ approaches 0 as well. This is done by finding conditions (epsilon and delta) that verify this behavior.

The aim is to ensure that no matter how close we want $x^3$ to be to 0 (determined by epsilon), there exists a region around 0 (determined by delta) within which $x^3$ behaves as desired.

This specific scenario teaches us how to handle and manipulate inequalities to maintain control over the function's behavior for small values of $x$.

Cube Root Relation

The cube root plays a pivotal role in relating $\varepsilon$ and $\delta$ in the limit problem of $x^3$ as $x$ approaches 0. To demonstrate that $x^3$ approaches 0, we need to express $|x^3| < \varepsilon$ using $|x| < \delta$.

Start by recognizing that $|x^3| = |x|^3$.
To relate it to $\varepsilon$, solve the inequality $|x|^3 < \varepsilon$ for $|x|$.
Taking the cube root on both sides gives: $|x| < \varepsilon^{1/3}$.

This means we can choose $\delta = \varepsilon^{1/3}$ to satisfy the condition. The use of the cube root is essential in finding a relationship between the small changes in $x$ and ensuring the outcome, $x^3$, remains bounded by any desired accuracy.

Verification of Delta Condition

Verification of the delta condition is the final step in applying the epsilon-delta definition to prove a limit. Once you have expressed $\delta$ in terms of $\varepsilon$, you must ensure that the conditions hold true throughout:

Suppose we choose $\delta = \varepsilon^{1/3}$.
We assume $|x| < \delta$.
Substituting this assumption back into our inequality $|x|^3 < \varepsilon$, we verify: $|x|^3 < \varepsilon$.

This process confirms the initial selections of $\delta$ and demonstrates logically that the original statement $\lim _{x \rightarrow 0} x^{3}=0$ holds true under the epsilon-delta definition. This verification provides the confidence that the relationship we established is consistent and indeed covers all intended scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$\lim _{x \rightarrow 0} x^{3}=0$$

Short Answer

Step by step solution

Understanding the Statement

Express |x^3| < 蔚

Choose 未 in terms of 蔚

Verification

Key Concepts

Epsilon-Delta Definition

Limit as x Approaches 0

Cube Root Relation

Verification of Delta Condition

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Statistics

Decision Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.