 In this video, I want to demonstrate how one can compute the limit of a difference quotient that involves a square root expression. So notice here that we have the limit as t goes to zero of the square root of t squared plus nine minus three over t squared. If we just were to naively plug in t equals zero because after all with a lot of these limit calculations, if you can just plug in the number zero in for the variable that often does the trick. So what happens here is you'll see that you get the square root of zero squared plus nine minus three over zero squared. The denominator, you can now see a problem there, zero squared becomes zero. But just because the denominator goes to zero doesn't mean there's no hope. In the numerator, you're going to get the square root of nine minus three, which the square root of nine of course is a three. We see the numerator is going to become zero over zero. So actually there is hope because we have this indeterminate form zero over zero. Zero over zero suggests to us is that we need more information to compute this limit. In fact, we probably need to simplify the function in some way or another. So how does one do that? Well, with you have a square root expression in your difference quotient. My suggestion to you is we want to rationalize the numerator. Now in previous algebra classes, you were probably taught a technique about rationalizing the denominator. And there was such insistence placed upon rationalizing the denominator that you felt like you should always rationalize the denominator as if it was always to the benefit of mankind. Frankly speaking, rationalize the denominator, although it's an important technique is very oversold that the reason we rationalize denominators so we can add fractions and involve square roots and other rational, other radical expressions, but it's not a silver bullet that solves all problems. In fact, in this situation, it'll be much more preferable to have an irrational denominator than an irrational numerator. So we're going to rationalize the numerator in this situation. So what that means again is you're going to take this expression with two terms. You have your first term right here, which is the square root. And then you have the second term, which in this case is just the three. You're going to switch the sign of this expression. So we have the square root of T squared plus nine. And then we're going to take plus three. We're going to times the top of the fraction by that. But to make sure that the proportion is retained, we need to multiply the denominator by it as well. So we get the square root of T squared plus nine plus three. Like so next, what we're going to do is we're going to multiply out the numerator. That's the whole point of rationalize the numerator. We have to multiply it out. And so when you do that, you're going to go through a classic foil. So you take the first outside, inside and last, right? So do all those possible multiplications there. When you take the square root of T squared plus nine times by itself, you're going to get the square root of T squared plus nine quantity squared. Then we're going to get the square root times three. So we get three times the square root. Then we're going to get negative three times that same square root. And then last, we get negative three times the positive three, which is a minus nine. So that's what happens to the numerator when you multiply it out. When it comes to the denominator, I have a nice little trick for you. Don't multiply out denominators. Again, in previous algebra classes, you might have been taught to do something like that, or you just did it because you were supposed to even know what I told you. And this is all a mistake. There is rarely ever a benefit to multiply out a denominator. We like denominators to be factored. For example, when it comes to simplifying fractions with the numerators factored, denominators factored, we can cancel the common divisors on top and bottom. We can simplify it when it's factored. So keep the denominator factored. When it comes to recognizing the domain of a rational function, it's much easier to do that when the denominators factored. Leave denominators factored. It is a fool's errand to multiply out a denominator. So do not do it. So I suggest everyone to make the unbreakable vow, pull out your wands, hold each other's hands, and make that unbreakable vow that I will not multiply a denominator. Don't do it, it's not gonna be good for you. It's the numerator that needs to be expanded and simplified. Now, some things you'll notice. You have three times the square root of t square plus nine. You have negative three times the square root of t square plus nine. Those are like terms, and when you combine them together, they actually cancel each other out. This is the main idea behind multiplying by this conjugate. This difference of the negative and the plus were playing off of the difference of squares factorization. a squared minus b squared equals a minus b and a plus b. So if you start with a minus b, if you times it by its conjugate a plus b, you'll actually get this difference of squares that in the middle terms cancel each other out. And so also notice, you're gonna get an a squared minus b squared. So we got the three squared versus nine, and then you take the square root of t squared plus nine. That should be a square there, sorry. Take the square root of t squared plus nine's quantity squared. It'll cancel out the square root, and we're left with then in the numerator, a t squared plus nine minus nine. That's how wonderful it is. Then you have your t squared on the bottom times the square root of t squared plus nine plus three as t approaches zero. Now, after we took over the middle terms from the foil and we squared the square root, we now have a t squared plus nine minus nine. The plus and minus nine cancel each other as well. And we end up with the limit of t squared over t squared times the square root of t squared plus nine. Plus three, excuse me, as t goes to zero. Now you see the benefit of not distributing the t squared in the denominator. We have a t squared in the numerator. We have a t squared in the denominator. These common divisors cancel out, and now we have a simplified rational expression for which it would then look like the limit as t approaches zero. Everything canceled out of the numerator, which means a one is left behind. In the denominator, we have this t squared, a square root of t squared plus nine, and then there's a plus three. It's like the thing we start off with, but now it's a plus three instead of a minus three. You'll notice now that as t goes to zero, we don't have a problem of just plugging in t equals zero anymore. The numerator will just be one, no big deal. The denominator is going to get the square root of nine plus three. Square root of nine, as we already identified, was three. So you get three plus three, which is one sixth. So because we're able to simplify the difference quotient, we then were able to evaluate the limit as t went to zero when you see that the limit in this case is one over six. So if ever you have a limit of a difference quotient involving a square root, I would recommend to rationalize the numerator by multiplying by the conjugate. With other radical expressions, you can also rationalize the numerator, although that technique is a little bit more complicated than what we saw in this video.