 So another type of indeterminate form we've run into is one of the form infinity over infinity. So this appears in problems like find the limit as x goes to infinity, 3x minus 5 over x squared plus 4x plus 7. And we might do our preliminary analysis as x gets large, then 3x minus 5 gets large, and x squared plus 4x plus 7 also gets very large, and so it appears that this rational expression goes to infinity over infinity, but infinity is not a number. You cannot divide by infinity. Infinity again expresses the idea of a quantity that grows without bound. You cannot divide by a quantity that grows without bound. So what can you do? Well, this is another indeterminate form, and the thing to remember is that these indeterminate forms always require additional analysis, very often in the form of algebra. So what sort of algebra might we do here? Well, let's take a look at that expression. As x gets large, again we do our preliminary check that we do actually have an indeterminate form, and so we know that we have to do some additional analysis. And here's the starting point. Since we know how to find the limit as x goes to infinity of 1 over x to the n, because as x gets large without bound, this expression goes to zero, we can multiply numerator and denominator by the reciprocal of the highest power of x in the denominator, and we'll see the reason for that as we work our way through the problem. So here's our original expression. I want to multiply by 1 over x squared, that's the highest power of x in the denominator, and then I'll do a little bit of algebra. I'll do a little bit of algebra. I'll expand those two expressions. I'll do a little bit of algebra, 3x over x squared, x squared over x squared, 4x over x squared, all those simplify, and now I'm ready to take a limit. And as x goes to infinity, this goes to zero, this goes to zero, this goes to zero, that goes to zero, and that just stays one. And this is why we multiply it by the highest power of x in the denominator, because it guaranteed we'd have a constant left at the end of the problem. So that's going to go to one, and everything else is going to go to zero, and after all the dust settles, my expression, the evaluated limit, is going to be zero. And while this problem didn't ask us to do so, it's always a good idea to try and find a numerical support of our claim. So as x goes to infinity, well, x is a very large number, 5 billion. If x is 5 billion, what does this expression evaluate to, and if you get something that's a very small number that's very close to zero, that suggests that all the work that we've done is correct. There are in general a lot of steps from the start of the problem to the conclusion to the problem. And there's lots of places to get the algebra wrong, to mis-copy a number, to do something you shouldn't be doing, and it's always nice to have that little bit of numerical support that says, yeah, we're heading in the right direction.