Question

Answered step-by-step

Problem 55 Hard Difficulty

(a) Use numerical and graphical evidence to guess the value of the limit
$$ \lim_{x \to 1}\frac{x^3 - 1}{\sqrt{x} - 1} $$

(b) How close to 1 does $ x $ have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?

Video Answer

Solved by verified expert

Textbook Answer

Official textbook answer

Video by David Mccaslin

Numerade Educator

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

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

Video Transcript

So in this problem were asked to use numerical and graphical methods to show. But the limit as x goes to one Of this function, x cubed -1 Over the skirt of X -1 equals. So first of all, numerically let's make a table here of excess and f of X. Okay, we'll have our access go to one first from the left. So if you put .5 into this formula you get 2.987 0.75 in here you get 4.315 0.90. In here You have 5.281 inching our way towards towards one right, 0.93 Gives me 5.49 0.99 Gives me 5.9- five. 0.999. Give Me 5.99- five. All right, so this the jess let the limit As X approaches one from the left of our function Is six, doesn't it? All right. Now Let's go from the other way. Let's start at 1.1 At 6.782 and 1.65 At 6.501 1.01 Is 6.075. 1.0 No one said get closer and closer to one. 6.075. So this suggests that the limit As X approaches one from the right then right, I'm coming in closer from the right Of our function is also six. So since I have the same limit from the left and the right then I have the limit. It's not writing very neatly. Let me try that again Limit as X approaches one of x cubed -1 over The Square Root of X -1. This function is six numerically, isn't it? All right. Let's see it graphically. Now I look at it graphically, here's the graph Y equals execute -1 over the square root of X -1. And what do I see as I get closer and closer and closer to one here I can't have the value at one but I can't get closer and closer and you notice that the values are getting closer and closer and closer. 26 all the time. Okay so that means graphically we were able to confirm this as well. Now the next question we're asked is how close 21 his ex for us to be within 0.5 of a limit. Well let's look at the table we made here. Okay The limit is six then I want to be at 5.5 to 6.5 don't I? And I can see right there that I have to be at .93 Or 1.065. Right Okay so I have to be 0.93 So that's equal to X is less than or equal to 1.065. Right within that range. And so I'm only allowed to give one delta here. So you will say that delta is absolute value X -1 for this. Um, where This limit as x approaches one Execute -1 over the square root of X -1. All right, that was the value of this um, -1, modest the limit. Right is within 0.5 Annex. I shouldn't say equal to is within their All right. So for that to happen, that means X is in this range, I can only get one value of delta So that delta is going to be, it's either .07 or .065 so I need the smaller one 0.65 is the answer that I'm looking for.

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.

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

More Answers

Related Courses

Calculus: Early Transcendentals

Related Topics

Discussion

Top Calculus 1 / AB Educators

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Michael Jacobsen

Calculus 1 / AB Courses

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

James Stewart

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Michael Jacobsen

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

Calculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Michael Jacobsen

James StewartCalculus: Early Transcendentals

Anna Marie Vagnozzi

Heather Zimmers

Kristen Karbon

Michael Jacobsen

James Stewart

Calculus: Early Transcendentals