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

If $ H $ is the Heaviside function defined in Exa…

09:57

Question

Answered step-by-step

Problem 37 Medium Difficulty

Prove that $ \displaystyle \lim_{x \to a} \sqrt{x} = \sqrt{a} $ if $ a > 0 $.

$ \biggl[Hint: \text{Use $ | \sqrt{x} - \sqrt{a} | = \frac{| x - a |}{\sqrt{x} + \sqrt{a}} $}. \biggr] $


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 4

The Precise Definition of a Limit

Related Topics

Limits

Derivatives

Discussion

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

Missouri State University

Heather Zimmers

Oregon State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

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

01:12

For $a>0$, use the iden…

00:58

$$\text { Prove that } \li…

01:20

Calculate.
$$\lim _{x \…

01:44

Show that
$$\lim _{x \…

01:10

Use the fact that $|x|=\sq…

03:28

Find the limit. Use l'…

0:00

Find the limit or show tha…

0:00

Find the limit or show tha…

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

Video Transcript

This is problem number thirty seven of the Stuart Calculus eighth edition section two point four. Prove that the limit as expert as a of the function square root of X is equal to the square root of a if a better than zero hint. Use the following identity, and we will see where this comes into play. But let's first recall Thie Absalon. Delta definition would limit for any Absalon is greater than zero. There's a delta graded in zero. You should find a delta greater than zero such that if he absolutely, with the difference between Max and A is this in Delta didn't have some value. The difference between f l is less than epsilon. And we can really write this in terms of what we have in our problem. A bean, eh? In this case, less than daughters. So that is about the same about our function here is going to be squared of X finest Lim score today his Listen, Absalom. And then here we see where the hint will come in because right now we have, ah, suitability of a difference of square roots that we don't really know how to work with. But using this to our advantage, we're going to transform this into ah, or re write the whole thing again. This is Listen, Absalon, but we replaced this, using our hint to make it into this. It's absolutely of the different Kleenex. And A is school out of X plus squared of a which is listen up, Salon. We're gonna make an observation. Ah, we know that the square root of X dysfunction eyes Ah, a positive value. Andi, If we hey simply remove it, we can stay true to the inequality. Ah, and have the comparison to Absalon. Just be the function here divided by the square today on this maintains the inequality. And that's an assumption that we made and such that we can solve for the difference between X and A That's a valley of that is going to be scared of a times Absalon. And now that we made this distinction, we can see the adults that I should be a good choice for. Delta should be that Aah! Delta is equal to escort of eight times Absalon on. We can show directly how this will prove this limit. We're going to take Ah, this exact step here this term this term here It's creative experience saying over score of X, Let's score today. Ah, we see that this it should be less than Epsilon because this is the case on. Then we make the substitution that absolutely of the difference between ixnay over just the skirt of a that still holds true. Ah, this limiting value. We chose to be Delta. So that's sort of a ah long. We will take this term actually which is a squirt of a it's delta squad of eight times up, absolute over this court of aging which we see cancels out to be Absalon on that is consistent with our definition. Therefore, this limit is expert is a of the functions Court of X is equal to the square today.

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
192
Hosted by: Ay?Enur Çal???R
Math (Algebra 2 & AP Calculus AB) with Yovanny
Arrow icon
Participants icon
82
Hosted by: Alonso M
See More

Related Topics

Limits

Derivatives

Top Calculus 1 / AB Educators
Catherine Ross

Missouri State University

Heather Zimmers

Oregon State University

Kayleah Tsai

Harvey Mudd College

Kristen Karbon

University of Michigan - Ann Arbor

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

01:12

For $a>0$, use the identity. $$ |\sqrt{x}-\sqrt{a}|=|\sqrt{x}-\sqrt{a}| \cdot \…

00:58

$$\text { Prove that } \lim _{x \rightarrow 0^{+}} \sqrt{x}=0$$

01:20

Calculate. $$\lim _{x \rightarrow 0} \frac{\sqrt{a+x}-\sqrt{a-x}}{x}$$

01:44

Show that $$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-x\right)=0$$ Hi…

01:10

Use the fact that $|x|=\sqrt{x^{2}}$ to show that $\lim _{x \rightarrow 0}|x|=0…

03:28

Find the limit. Use l'Hopital's rule if it applies. $$\lim _{x \rightarrow a} \…

0:00

Find the limit or show that it does not exist. $ \displaystyle \lim_{x \to \…

0:00

Find the limit or show that it does not exist. $ \displaystyle \lim_{x \to …

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