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Find the limit or show that it does not exist. …

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Problem 29 Medium Difficulty

Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to \infty} \left(\sqrt{x^2 + ax} - \sqrt{x^2 + bx} \right) $


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03:45

Daniel Jaimes

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Related Topics

Limits

Derivatives

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Top Calculus 1 / AB Educators
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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.

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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
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
Problem 78
Problem 79
Problem 80
Problem 81

Video Transcript

Okay, when you get, when you plug infinity and you get infinity minus infinity, which is indeterminant. Okay, so we're going to do something and I've tried other some things, but here's what I'm gonna do this time manufacture out a square root of X. Now I'm going to factor it all the way out of those square roots. So I have the square defects times square root of X plus a minus squared of X plus B. Okay, So I still have infinity minus infinity and then also at times infinity. So that doesn't seem better. But it is what I'm gonna do is I'm going to multiply the top and bottom of this by the conjugate. Okay. And here's what that is. If I am let's say um uh C plus D. And I multiply it by its conjugate, which is c minus D. Then all I do is square the first one. Put a minus sign and square the second one Difference of two Squares. That's what cons gets do. So I'm going to multiply by the square root of X plus A plus the square root of X plus B on the top and bottom. So put it over one. So it has the bottom. Now I'm going to multiply by X plus A plus the square root of X plus B. Okay, and that's a trick that you use whenever you have a square root plus or minus something. If one or the other or both are square root, then the continent almost always will get you out of the mess. Okay, so you get the square root square the first thing put a minus sign square the second thing. Okay, that's all over squared of X plus A plus the square of X plus b. Okay so the x minus X. Those cancel. So now I have limit as X goes to infinity squared of X times A minus B over the square root of X plus A plus the square root of X plus beat. Well, okay, so now I get infinity plus infinity on the bottom and infinity on the top. Okay, so that's still undefined. So we're gonna think of something else to do. So then what I did was I remember that I factored the square root of X out so maybe I should put it back in there. Okay, so what I did next was I multiply the top and bottom by one over the square root of X. So on the top those cancel and I get a -1. And on the bottom I get squared of X plus A. Over the square root of X plus the square root of X plus b. Over the square root of X. So that can go in there and separate. Second limit as X approaches A A minus B Over Oh limit as X approaches infinity. Okay, sorry, square root of X or square or X over X. Which is one plus A over X Plus one Plus B over X. Alright. Yeah, now I can take the limit because the top is a number is A minus B and the bottom is well, as X goes to infinity, that goes to zero and that goes to zero, so it's the square to one plus the square to one, so A minus B over two. That's what I say. The limit is.

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Top Calculus 1 / AB Educators
Grace He

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Heather Zimmers

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Caleb Elmore

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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.

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