Download the App!

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

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

Problem

Let $ f(x) = [ \cos x ] $, $ -\pi \le x \le \pi $…

04:08

Question

Answered step-by-step

Problem 53 Hard Difficulty

(a) If the symbol [] I denotes the greatest integer function defined in Example 10 , evaluate
(i) $\lim _{x \rightarrow-2^{+}}[x]$
(ii) $\lim _{x \rightarrow-2}[x]$
(iii) $\lim _{x \rightarrow-24}[x]$
(b) If $n$ is an inleger, evaluale
(i) $\lim _{x \rightarrow \infty}[x]$
(iii) $\lim _{x \rightarrow \infty}[x]$
(c) For what values of $a$ dacs lim. $. . .[\mathrm{xl}]$ caist?


Video Answer

Solved by verified expert

preview
Numerade Logo

This problem has been solved!

Try Numerade free for 7 days

Daniel Jaimes
Numerade Educator

Like

Report

Textbook Answer

Official textbook answer

Video by Daniel Jaimes

Numerade Educator

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

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Related Topics

Limits

Derivatives

Discussion

You must be signed in to discuss.
Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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.

Join Course
Recommended Videos

06:05

(a) If the symbol [[ ]] de…

01:16

(a) If the symbol Il denot…

01:17

(a) If the symbol I 1 l de…

04:49

(a) If the symbol II denot…

04:53

(a) If the symbol denotes …

07:28

Which of the following lim…

03:25

$$
\begin{aligned}
&…

01:21

If $[x]$ denotes the great…

Watch More Solved Questions in Chapter 2

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66

Video Transcript

is problem number 53 of the Stuart Calculus eighth edition. Section 2.3 Party of the symbol brackets notes the greatest interject function defined an example. 10. Evaluate the following limits. So let's take a look and refresh our understanding of the greatest interest your function. Here we have an example where between say, 10 and one. The value of the function is zero because it is the greatest integer up to that point, moving from left to right on the function. Once the function reaches the value of one, the following integer The function takes on that value of that integer until it reaches the next integer to. So this seems like a step function where we see that the greatest integer is represented in the function value part one. The party. What is the limit as X approaches negative two from the right of this greatest interviewer function Mhm. As we see, we look at where negative two is. We follow the function from the right towards negative too. And we see that we are still along the line of negative two. And so this limit that we're approaching is equal to negative two. What is the limit as we approach negative, too, for this function, well, this has to do with the lift, the limit from the left and the right, as we understand when there is no specification, whether it's from the left or the right. Then we determined the limit. Based on both of the limits, we saw that from the right to the limit as you first, negative two is negative. Two. As you approach negative two from the left, we see that limit is equal to negative three. So because the limit from the left and the right do not agree at negative two, we say that the limit does not exist. Part three. The limited expertise. Negative 2.4. For this integer function, the 2.4 falls somewhere between negative two and three. If we go down to the function, notice that it is a constant value, so negative 2.4. Here we have to understand where the limit is coming from the left and the right notice that no matter where you are, between these two indicators negative three negative two. The limit is always going to be equal to negative three because the function stays constant from both the left and the right towards any point between those two integers. So the limit here does exist, and it's equal to negative three. Puppy. If N is an integer, evaluate the limit as X approaches end from the left. Let's take a look here. Let's pick a value N for an example here and is one, Let's say, And if we're approaching one from the left, we see that the limit of that function in zero. Let's take the next value to as we approach and is equal to two. From the left, we see that the value of function as one. So we see that as you approach this function, you approach a value event from the left, you get a value that's one less than the value of N. And so we conclude that the limit is always going to give you one less than the value of N as the limit from the other direction approaching end from the right, let's say we choose N is equal to one from the right, we see that the limit is equal to one. Let's say we chose and is equal to negative one. If we approach and equals negative one from the right. We are at negative one, and this is characteristic of the entire function. That the limit, as you approach end from the right for this function is equal to the value and itself. Now, for part C for what values of aid is the limit, as X approaches in of the greatest interest function exists well, we noticed that the limit did not exist when X equals negative two. Here we see that every time there's a jump discontinuity in the function, the limit does not exist, and we see that the jump occurs at every time that there's a new integer negative to negative 101 and so on. So the limit as the expertise eight exists for all values. Besides a is an integer, as we see here, when a is an integer imagining and is an integral here. Notice that the limit from the left is different from the limit from the right, always but as we saw for any value between the two integers, such as in the example, Part three of a, you are always going to get a limit that exists because the value is constant. And when you approach the any point from the left and the right, you will always get that same constant value. So we say that the values no way that satisfied this in it are all numbers all real except a equals unreal except and teachers, and that concludes our solution.

Get More Help with this Textbook
James Stewart

Calculus: Early Transcendentals

View More Answers From This Book

Find Another Textbook

Study Groups
Study with other students and unlock Numerade solutions for free.
Math (Geometry, Algebra I and II) with Nancy
Arrow icon
Participants icon
142
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
68
Hosted by: Alonso M
See More

Related Topics

Limits

Derivatives

Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Caleb Elmore

Baylor University

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

04:40

Limits - Intro

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.

Video Thumbnail

04:40

Derivatives - Intro

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.

Join Course
Recommended Videos

06:05

(a) If the symbol [[ ]] denotes the greatest integer function defined in Exampl…

01:16

(a) If the symbol Il denotes the greatest integer function defined in Example 1…

01:17

(a) If the symbol I 1 l denotes the greatest integer function defined in Exampl…

04:49

(a) If the symbol II denotes the greatest integer function defined in Example 8…

04:53

(a) If the symbol denotes the greates integer function defined in this example …

07:28

Which of the following limits exist? (where [.] indicates greatest integer func…

03:25

$$ \begin{aligned} &\lim _{x \rightarrow 1} \frac{x \sin (x-[x])}{x-1}, \text {…

01:21

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value…
Additional Mathematics Questions

03:18

An elevator descend into a mine shaft at the rate of 6m/min. The descent sta…

01:03

What is the place value of 4 in 874569036 in the Indian system and internati…

01:09

Which of the following options represent consecutive numbers?
1. 1,23,45,…

02:28

4. Evaluate(i) 15625 x (-2) + (-15625) x 98 show me steps

00:59

Find the coefficient of X in the expansion of (X + 1)^3

05:00

If the positions of the first and the third digit within each number are int…

04:13

Evaluate whole root 5-2 root 6 + whole root 10 - 2 root 21

01:09

8 years ago there were 5 members in the Arthur's family and then the av…

01:06

A bag contains 7 green and 5 black balls. Three balls are drawn one after th…

01:15

Find the value of p for which x =-2, y=-1 is a solution of the linear equati…

Add To Playlist

Hmmm, doesn't seem like you have any playlists. Please add your first playlist.

Create a New Playlist

`

Share Question

Copy Link

OR

Enter Friends' Emails

Report Question

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started