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Find the limit.$$\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)$$

$$\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)=\frac{1}{6}$$

Calculus 1 / AB

Chapter 1

FUNCTIONS AND LIMITS

Section 6

Limits Involving Infinity

Functions

Limits

Continuous Functions

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so is getting more complicated as we go forward. But it is not. Anything difficult is toe requires a little bit of smartness from you, and you can you can sail through carefully and peacefully. Right now. This one is not a rational function. It is just a list, a linear function, right. Rational function is a function that you can have a photo like. Experts two or X minus three. Just irrational function. But it's when there's nothing over right. It's just a linear function or they see whole function, right? So how do we do something like that? Well, we can still force away through they making it rational function if we decided to use congregate certain. So this is he just multiply it by the congregate. So once you do that, then you make it rational. Now, now, when you multiply the top one, you know that is gonna be the first term square. The first time squared is just that's what records square root. So when you square, it is just gonna be the number of yourself, then minus the second term square. This is the second term. So when you square, it is gonna be nine x squared, right? And then over that right. You have this one and they taking the limit as X approaches infinity off such a thing. Right now we know that this is night positive. Nine x creators is negative. Nine x were so they're gonna subtract out. So what is gonna go left here is just x here, right? And this is gonna be a nine x squared plus X I m plus three x and taking the limit as X goes to infinity right now, what is the highest power of X? The highest power of X is not X squared because of the square root. Right, So there's power of aces this X you right? So the highest pile of eggs X there, then when you can do is divide each and every side but the numerator and the denominator by X. Right. So there's gonna be X over eggs, and then the denominator is also gonna be That's very tough. Nine X squared plus X right in close, three explosive over X. Like that. Right? So then this one is gonna be one, right, cause the top is gonna cancel each other. And in here we can we can share, right? We're gonna share. So we share that this and then this is also gonna be so this is gonna take eggs, and this is gonna take X. We just share it amongst theist enumerators, and this cancels this one. So our problem here is this. Well, who was the last tutorial? You know how to kit this X inside the square. Right? So I'm just gonna do it straight away, So there's gonna be limit, as except which is that it's gonna meet that first is gonna be, uh, this do this. The new mayor is gonna be one. Right? So that is what you have that one for. And then here, this is gonna be a nine x squared plus eggs. We're gonna push this exterior into the X squared by squaring it right? Remember, we do that in the last tutorial. So this is gonna be, uh, this is gonna be here. This is gonna be, uh, exclude as well. Right? And then close. My three left, Miss my three here. Right. So when I also share this X, where to each of these two terms, then I'm gonna have one over square root off nine X squared over X squared, then plus X over X squared, then plus three. So finding them having limits as X approaches Infinity one over. Swim route off. Now, this was gonna cancel this way, so I'm gonna have nine. Then this one is gonna cancel one of these is gonna be one over X, then plus three. So when I send X to infinity, then you can see that this one is gonna eventually go to zero. Right? So what I'm gonna have is one over a squared off nine plus three. What is good if I screwed of nine is to me. And three plus three is six services, 1/6.

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