 In this video I wanna present a technique so we can simplify limits of difference quotients that involve reciprocal or rational functions. Now you'll notice in this setting here we have the limit as h approaches zero of one over x plus h squared minus one over x squared all over h. If you were just to plug in h equals zero in this setting because that's a very smart thing to do when you have a limit, just plug it in and see what happens. You can end up with a one over, since h is going to zero, you're gonna get one over x squared minus one over x squared all over zero. Now division by zero is usually a red flag. That's why I'm writing this in blood right now. And division by zero is a bad thing, but you'll notice in the numerator you also get zero. And so zero divided by zero isn't actually so bad. This is what we call an indeterminate form because with the information we have right now we cannot determine whether the limit exists or not. And if it does exist, is it an infinite number or is it a finite number? We don't know, we have to investigate it a little bit further. And in fact, because we get zero over zero, it means that there's probably some type of algebraic simplification we can do to the function in order to calculate its limit. So let's try to do that. So when you have our function, let me just write it down here again. You're gonna have an h in the denominator. You're gonna have a one over x plus h squared in the numerator minus a one over x squared, like so. So this right here is what I often refer to as a compounded fraction. Some people call them nested fractions. Some people call them complex fractions. I don't really care for the term complex fractions because that sometimes implies this has something to do with complex numbers and it's like, no, we really just need to have fractions inside of fractions. So a compounded fraction or a nested fraction is a better term in there. So we have fractions inside of fractions. And so what you wanna do is identify what are the little baby fractions, you know, the right here and then you have your big mommy fraction, you know. So this is the big fraction bar, that green fraction bar we're not gonna make go away. But the little fractions, the baby fractions, we can take them out of the problem here. And to do that, we wanna identify what is the least common denominator of all babies? So if you look at the first baby, its denominator is x plus h squared. So in terms of our least common denominator, the least common denominator is gonna need to have an x plus h squared in it because the first fraction has that. And then the second baby fraction, one over x squared, it also has its denominator as x squared. That's not, there's nothing alike, no common divisors of x squared and x plus h squared in general. So we have to add that also to our least common denominator. So what we're gonna do is we're gonna times the top of the mother fraction by the LCD and times the bottom also by that LCD to keep things proportional. So I'm gonna times the numerator by x squared times x squared, x plus h squared. And we have to do that in the denominator as well. Okay, now in the numerator because the numerator of the original fraction is a difference, we can distribute this onto both terms. That's really what we want to happen. So we're gonna get the limit as h goes to zero. We're gonna get x squared over x plus h squared all over x plus h squared. Subtract from that x squared times x plus h squared over x squared. Now in the denominator of the mother fraction of h times x squared times x plus h squared. Let me give you a tip. When it comes to a denominator of a fraction, leave it factored. We do not multiply out denominators. It is a fool's errand. It doesn't provide you any benefit. It just adds extra workload that you're gonna have to undo anyways and then we have to factor it. So we're gonna leave the denominator factored. h times x squared times x plus h quantity squared. So how can we simplify things? When you look at the first baby fraction, you'll notice it has an x plus h squared over an x plus h squared, those cancel out. And that'll leave behind just an x squared in that numerator as h is going to zero right here. When you look at the second baby fraction, it has an x squared on top and x squared on bottom that cancels out and we get an x plus h quantity squared. And then because we made a blood oath not to multiply out denominators, we still have an h times x squared times x plus h squared. So we must fight the temptation to multiply out the denominator there. On the other hand, the numerator we should expand, expand as much as we can in order to simplify it. So notice you have that x plus h quantity squared. You can foil that out. You're gonna get an x squared plus two x h plus h squared. Be careful that the negative sign attached to the x plus h squared will distribute onto all these pieces. Actually, write it out, distribute it if you needed to in case you forget. And so you'll notice now there's an x squared over here. There's a negative x squared right here. Those terms are gonna cancel each other out. And so who's left in the numerator? You're gonna have this negative two x h, remember I just drew that negative sign. You're gonna have a negative h squared and this sits above h x squared and x plus h squared. Like so as h goes to zero. Now our whole goal, and when it comes to these difference quotients, your goal is you wanna get rid of the h in the denominator. H stands in this case for hate. We hate the h in the denominator and so we must get rid of it. But we can't just hire some hit man to come take out the h. We have to do it algebraically. And so in order to cancel the factor of h in the denominator, we need a factor of h numerator, which now, once I've looked at everything that's canceled, I'm now left with a negative two x h minus an h squared. Everyone in the numerator is divisible by h. We can factor it out to get h times negative two minus h sitting above our denominator h x squared times x plus h squared as h goes to zero. Notice now the factor of h cancels on top and bottom. And so now our difference quotient is simplified. I want you to be aware that as we've gone through this process, everything we've done thus far has been purely algebraic. No calculus has actually been involved in this process other than what the heck does a limit mean. All right, this is purely just an algebraic simplification. This is something you see often in calculus that when it comes to a calculus problem, there's probably only like one, maybe two lines of work that's genuinely calculus. Everything else is pre-calculus. It's for this reason why we need to be excellent at pre-calculus in order to do these calculus problems. Without pre-calculus, we cannot do calculus. The calculus part's not the hard part. It's the pre-calculus, frankly speaking. Now in this setting, you'll notice that if we set h equal to zero now, there'll be no longer division by zero. And so now we're ready to compute the limit. Setting h equal to zero, we get negative two x minus zero above x squared, because we're setting h to zero. We're not doing anything with the x. x squared times x plus zero squared. So if we simplify this, the numerator becomes a negative two x, the denominator becomes an x squared times an x squared, which that would become a negative two x over x to the fourth. We can cancel the x on top with one of the x's on the bottom, and we see that this limit will simplify to be negative two over x cubed, like so. For which the original expression depended on the variables h and x, and so as h goes to zero, we see that the expression will become negative two over x cubed.