Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Prove the statement using the $ \varepsilon $, $ \delta $ definition of a limit.

$ \displaystyle \lim_{x \to 0} x^3 = 0 $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Daniel Jaimes

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 4

The Precise Definition of a Limit

Limits

Derivatives

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

01:59

Prove the statement using …

04:47

0:00

03:38

07:16

01:55

this problem. Number twenty six of this tour. Calculus s edition, Section two point four. Prove the statement using the Absalon Delta definition of eleven. Tell him it is experts. Zero x cubed. Is he called a zero? The Epsilon Delta definition of element states that this limit is true and is equal to zero as long as there's an absalon to greater than zero that for every Absalon greater than zero, you find a delta greater than zero such that if the value of the difference between X and a his lesson delta, then the asset value of the difference being the function and the limit l is less than Absalon. We'LL begin with the second inequality and specifically playing our function of interest are function here is X cubed minus still amid which is equal to zero must be. Listen, Absalon, if we simplify this just a bit, get the cube root. And if we take the cube root on both science, we should get this statement here. Be absolutely vexes. Listen, the cube root of Absalon working with this first inequality the absolutely of X minus A. In our case, a is zero. That's what X is approaching is less than Delta should be. Zero. And we if we simplify experience zero, it just gives this up. So I vexes. Listen, Delta. And then at this point, we can compare these two inequalities and see that an appropriate choice of Delta would be dealt is equal to the cube root of Absalon. And this relationship itself proves that there is a delta created zero for any absolute zero as long as it's a plan is greater than zero that satisfies all these inequalities and statements as satisfies the definition of a limit. The excellent if not the definition of a limit which proves our statement proves the limit exists and is he cool to zero?

View More Answers From This Book

Find Another Textbook

03:26

(4 pts ) For each year the populalion of flowers in @ garden is represented …

02:34

The graph of the rational function, g(x), is given to the right It is made u…

01:26

SPRECALC7 3.4.006,Mi;MYNOTLSASk YouR TexchERList 4M pos? Dle rto…

01:05

Which of the following planes is the equation for the plane with an x-interc…

01:58

QUESTIONMicroprocessor 68000 Motorola system consists of three (3- main …

02:27

Find the daily consumer surplus if the demand equation is given by P = ~0.1.…

02:48

Calculate a business's producer surplus if they sell an item at $ 15 , …

03:29

Hw22-double-integrals-over-general-regions: Problem 11Problem Value- poi…

12:13

Use the functions fand g in C[-1, 1] to find (f, g), Ilfll; Ilg l, and d(f, …

00:57

Identify the vertex by writing the equation in the standard form of a quadra…