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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) $

Answer

$\frac{a-b}{2}$

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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.