>> All right, let me show you a technique that is useful in this algebraic arena, another technique. The limit as X approaches infinity of this fraction. Now, look at the numerator and denominator. You'll notice that if, if this X gets really, really large and that X gets really, really large, the, the numerator actually is approaching infinity, the denominator is actually approaching infinity and infinity over infinity is an in determinant form. We cannot determine the limit in that form. So what we're going to have to do is change the form of this fraction. We're going to have to manipulate the fraction a little bit to create a situation where, and here's the target on this. You're trying to find a situation where the denominator is in a form where the limit of the denominator is some real number. That's what you're trying to do. And so, and the technique that we're going to use here is to divide out this X to get a cancellation opportunity with this X. And to do that we'll multiply through by 1 over X over 1 over X, a form of 1. Now 1 over X times the 2X is 2X over X. 1 over X times minus 1 is minus 1 over X. 1 over X times 3X, 3X over X, 1 over X times 2 is 2 over X. And then the, the Xs cancel here, here, here, here. So we have 2 minus 1 over X, 3 plus 2 over X in the denominator. And now oh happy day, we're in that, that nice condition that we're after because we can think of the limit as X approaches infinity. Now for this constant, now we can apply the limit idea to each of the terms that we see here. We could actually write that down, but generally, you won't do that. You probably will not write it down. You'll just consider each item in the fraction as sort of a separate event. The limit for 2 is 2. The limit of 1 over X as X goes to infinity, that fraction goes to 0. And the limit for the constant of 3 is 3 and the limit for this fraction as X gets really big becomes 0 as well. So we have this and therefore 2/3. So the limit as X approaches infinity for the original fraction turns out to be 2/3.