Limits at infinity using algebra | Limits | Differential Calculus | Khan Academy

Let's think about the limit of the square root of 100 plus x minus the square root of x as x approaches infinity. 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.

First, let's just try to think about it before we try to manipulate this algebraically in some way. So, what happens as x gets really, really, really large? As x approaches infinity, well, even though this hundred is a reasonably large number, as x gets really large—billion, trillion, trillion trillions, even larger than that— you can imagine that the hundred under the radical sign starts to matter a lot less.

As x approaches really, really large numbers, the square root of 100 plus x is going to be approximately the same thing as the square root of x. So, for really large x's, we can reason that the square root of 100 plus x is going to be approximately equal to the square root of x. In that reality, as we keep increasing x's, 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 algebraic manipulation to feel better about that instead of this kind of hand-wavy argument about the hundred not mattering as much when x gets really, really large.

Let me rewrite this expression and see if we can manipulate it in interesting ways. So, this is 100 plus x minus the square root of x. One thing that might jump out at you whenever you see one radical minus another radical is: 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. To not change the value of this expression, we can only multiply it by 1. 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.

Let's multiply this times the square root of 100 plus x plus the square root of x over the same thing, square root of 100 plus x plus the square root of x. Now notice this, of course, is exactly equal to 1. The reason why we like to multiply by conjugates is that we can take advantage of differences of squares.

This is going to be equal to—in our denominator—we're just going to have the square root of 100 plus x plus the square root of x. In our numerator, we have the square root of 100 plus x minus the square root of x times this thing, times the square root of 100 plus x plus the square root of x.

Now, right over here, we're essentially multiplying a plus b times a minus b, which will produce a difference of squares. This top part right over here is going to be equal to—let me do this in a different color—it's going to be equal to this thing squared minus that thing squared.

So, what's (100 plus x) squared? Well, that's just (100 plus x)(100 plus x), and what is (square root of x) squared? Well, that's just going to be x. So, we have minus x. We do see that this is starting to simplify nicely. All of that over the square root of 100 plus x plus the square root of x.

These x's, x minus x, will just be nothing. So we are left with a hundred over the square root of 100 plus x plus the square root of x. We could rewrite the original limit as the limit as x approaches infinity. Instead of this, we've just algebraically manipulated it to be this: the limit as x approaches infinity of 100 over the square root of 100 plus x plus the square root of x.

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 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.

That is going to approach zero, which is consistent with our original intuition.