 In a previous video, we learned that if two functions only differ from each other by one point in their domains, then the two functions will have the same limit. This becomes extremely useful in situations where you have a rational expression of some kind like the function you see on the screen here, 3h, 3 plus h squared minus 9 over h. In these settings, there might be values that make the denominator go to zero. Like in this case, if we were simply just to plug in h equals zero, we'd get a zero in the denominator, which is no good. Now, you'll notice if you were just simply to plug in h into the numerator as zero, you'd get a 3 plus zero squared minus 9. With a little bit of simplification, you end up with 3 squared minus 9 over zero. You get 9 minus zero over zero. You get zero over zero. And so this is actually a very common occurrence when you're working with limits. In particular, in this situation, we have what we call a difference quotient. This right here, the name is super clever. It's like calling a dog a fuzzy, panting, barking quadruped. We're just describing what we see right here. And so a difference quotient is exactly that. You'll see this difference in the numerator and then it is a divide by something. So it's a difference quotient, very clever name. But it turns out this difference quotient setting actually shows up a lot. And at the end of this chapter, we'll actually start learning about a concept known as the derivative, which will emphasize why the limit of a difference quotient is actually such an imperative idea for us to study here. Anyhow, when you get something like zero, even zero, excuse me, zero divided by zero, this suggests there's some type of hope that is the limit could still exist, right? The limit could be seven, it could be infinity, it could be zero, it could be one, it could be negative infinity, or it might not exist, right? There are some options that we don't know. And so this zero over zero is often referred to as an indeterminate form. And so that limit law we had mentioned earlier about if two functions differ only by one point they have the same, they still have the same limit. We can use that, that is, could we potentially simplify this rational function to cancel out the division by zero that's happening because of that factor of eight to the denominator. And the fact that this limit has the form zero zero suggests to us that yes, we algebraically can simplify this expression. And so let's proceed to do that. We're taking the limit as H approaches zero of this thing. The denominator is really not much to do, it's just this factor of H. But we can expand out the numerator because when you see things like three plus H squared, this means three plus H times three plus H. We can foil this thing out. We end up with nine plus six H plus H squared. That then goes inside of the numerator, nine plus six H plus H squared. And then you subtract nine from that. You'll now notice that the first thing, the three plus H squared when expanded, has a nine in it, which when subtracted the nine that's there, they cancel, in which case then the numerator simplifies to become six H plus H squared over H. And so looking at the numerator, we notice that, hey, everyone in the numerator is divisible by H. There's an H right here. If we factor out that H, we end up with the limit of H times six plus H, all over H, taking this limit as H goes to zero. For which we now identify there is a factor of H in the numerator. There is a factor of H in the denominator. And so we can cancel common divisors in the numerator and denominator. So this H cancels right here. Thus the expression would then simplify to the limit as H approaches zero of six plus H. And so this is the part we want to emphasize here. The function Y equals six plus H, that's a linear function. We have this original rational function right here, three plus H squared minus nine over H. Now these two functions right here, highlighted here in yellow, they differ from each other in only one point. They differ with each other at H equals zero. Because the rational expression is undefined when H equals zero because you get zero over zero. On the other hand, this linear expression six plus H is perfectly well defined when H equals zero. If we take the limit as H goes to zero, we end up with six plus zero, which just gives you six. And since the two functions only differ by one point, the limit of six plus H is the same as the limit of three plus H squared minus nine over H. And therefore, if we can simplify difference quotients algebraically, we can compute the limit by setting H equal to zero in the simplification when the original function didn't have that option.