Limit at infinity of a difference of functions

- Let's think about the limit of the square root of 100 + x - the square root of x, as x approaches infinity and I encourage you to pause this video and try to figure this out on your own. So I'm assuming you've had a go at it. So first let's just try to think about it, before we try to manipulate this algebraically, in some way. So what happens is x gets really really really really large, as x approaches, as x approaches infinity. While even though this 100 is a reasonably a reasonably large number, as x gets really large, Billion, trillion, trillion trillions. Even larger than that. Trillion trillion trillion trillions. You could imagine that the 100, under the radical sign, starts to matter a lot less. As x approaches really really large numbers, the square root of 100 + x is going to be approximately the same thing as the square root of x. So for really really large large x's, we can reason that the square root of 100 + x is going to be approximately equal to the square root of x and so in that reality, we are going to really really really large x's. In fact, there's nothing larger, where you can keep increasing x's, then these two things are going to be roughly equal to each other. So it's reasonable to believe, that the limit as x approaches infinity here, is going to be zero. You're subtracting this, from something that is pretty similar to that, but let's actually do some algrebraic manipulation to feel better about that, instead of this kind of hand-wavy argument about the 100 not mattering as much, when x gets really really really large and so let me re-write this expression. See if we can manipulate it in interesting ways. So this is 100 + x - x. So one thing that might jump out at you whenever you see one radical, minus another radical like this. Well maybe we can multiply by its conjugate and somehow get rid of the radicals, or at least transform the expression in some way, that might be a little more useful when we try to find the limit, as x approaches infinity. So let's just and obviously we can't just multiply it by anything arbitrary, in order to not change the value of this expression, we can only multiply it by one. So let's multiply it by a form of one, but a form of one that helps us, that is essentially made up of its conjugate. So let's multiply this let's multiply this times the square root of 100 + x + the square root of x, over the same thing. The square root of 100 + x + the square root of x. Now notice this, of course is exactly equal to one and the reason why we like to multiply by conjugates, is that we can take advantage of differences of squares. So this is going to be equal to and our denominator, we're just going to have we're just going to have the square root of 100. Let me write it this way actually. 100 + x + the square root of x and our then our numberator, we have the square root of 100 + x, minus the square root of x times this thing. Times square root of 100 + x + the square root of x. Now right over here, we're essentially multiplying A + B times A - B, will produce a difference of squares. So this is going to be equal to this top part right over here, is going to be equal to is going to be equal to this, let me do this in a different color. It's going to be equal to this thing squared minus, minus, this thing, minus that thing squared. So what's 100 + x squared? Well that's just 100 + x, 100 + x and then what square root of x squared? Well that's just going to be x. So minus x and we do see that this is starting to simplify nicely. All of that, over the square root of 100 + x + the square root of x and these x's x - x will just be nothing and so we are left with 100 over the square root of 100 + x + the square root of x. So we could re-write the original limit, as the limit, the limit as x approaches infinity. Instead of this, we just algebraically manipulated it, to be this. So the limit as x approaches infinity of 100 over the square root of 100 + x + the square root of x and now it becomes much clearer. We have a fixed numerator. This numerator just stays at 100, but our denominator right over here, is just going to be, it's just going to keep increasing. It's going to be unbounded. So if you're just increasing this denominator, while you keep the numerator fixed, you essentially have a fixed numerator, with an ever-increasing, or a super large, or an infinitely large denominator. So that is going to approach, that is going to approach zero, which is consistent with our original intuition.