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 1}\frac{2 + 4x}{3} = 2 $
Video Answer
Solved by verified expert
This problem has been solved!
Try Numerade free for 7 days
Like
Report
Official textbook answer
Video by David Mccaslin
Numerade Educator
This textbook answer is only visible when subscribed! Please subscribe to view the answer
03:23
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 4
The Precise Definition of a Limit
Limits
Derivatives
Oregon State University
Harvey Mudd College
University of Michigan - Ann Arbor
Boston College
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.
0:00
Prove the statement using …
04:47
02:28
02:33
04:22
08:05
03:38
03:51
$19-32$ Prove the statemen…
So in this problem, were asked to prove That the limit as X approaches one of two plus four, X over three equals two. Using the epsilon delta definition. Well, the epsilon delta definition says that if x minus one, so that's in delta. And that implies that f of x minus L is less than epsilon. Where L is the limit. All right. So, We're gonna let ε greater than zero be given then. We're gonna choose delta Equal to 3/4 times epsilon. Mhm. Okay. So then we have we'll see this was the x minus a. Sorry? So a is one F of X is two plus four X over three, and L is too All right. So then I have observations X minus one. Less than delta implies. What? That F of x minus L. Well, that's the value of two plus four. X over three minus two. Okay. And this is the absolute value of two plus four. X minus six. All over three. Well, So then this is four X -4. Yeah, civilian over three. Which is absolutely of 4/3, which that's already positive. So we can just factor it out straight like it is Comes up the value of X -1. Right, Okay. But X -1 is less than Delta. So this is less than 4/3 Delta. And Delta was three or 4 ε. This is 4/3 times 3/4 epsilon. And these cancel. So this is epsilon. So I mean fx minus l is less than epsilon. So therefore we have no the limit As X approaches one of two plus four, X over three equals two. Bye, epsilon delta definition. Yeah. And
View More Answers From This Book
Find Another Textbook
05:49
a. How much force does an 80 kg astronaut exert on his chair while sitting a…
06:43
A newspaper reports that the governor’s approval rating stands at 65%. The a…
00:44
Here are histograms for four manufactured sets of numbers. The histograms lo…
10:27
A survey of major universities asked what percentage of incoming freshmen us…
01:36
Jasper’s Grocery places an order for Monster every 3 weeks. Once the order i…
05:09
Brash Corporation initiated a new corporate strategy that fixes its annual d…
05:24
According to the National Education Association (NEA), teachers generally sp…
01:03
Martinez Auto Supplies has retail stores located in eight cities in Californ…
06:40
Annual revenue for Corning Supplies grew by 5.5% in 2007; 1.1% in 2008; −3.5…