 Suppose we have two functions f and g such that the limit as x approaches a of f of x turns out to be zero and the limit as x approaches a of g of x turns out to be plus or minus infinity. What would happen then if we take the limit as x approaches a of f of x times g of x? Well, then we would expect that the limit is going to look something like zero times infinity, which what is that? This is another example of an indeterminate form. It's actually a product indeterminate form and we can use L'Hopital's rule to help us resolve this indeterminate form as well. Now, if we're being honest, L'Hopital's rule only applies to the indeterminate forms zero over zero and infinity over infinity. It doesn't apply to the indeterminate form zero times infinity directly, but it turns out that any product can actually be turned into a quotient. If you have the product f times g, that's the same thing as f divided by the reciprocal of g. And so you can turn products into quotients for which then in that case, if you push g to the denominator, this would look like zero over one over infinity, which would then just become zero over zero, which is perfectly acceptable. On the other hand, if you pushed f into the denominator, you'd have g over one over f. The limit here would look like infinity over one over zero, which is going to be infinity over infinity. So it doesn't matter who gets bumped to the denominator, push the simpler of the two functions that involved into the denominator because you're going to have to take the derivative of them. And then we can turn these product indeterminate forms into quotient indeterminate forms for which L'Hopital's rule applies. So consider, for example, the limit as x approaches zero from the right of x times the natural log of x. If we plug in x equals zero here, we're going to get zero times the natural log of zero from the right. The direction does matter here because in terms of the domain of the natural log, we can't approach zero from the left. But in this situation, you get zero times negative infinity and the sign of the infinite doesn't really matter because zero doesn't care about signs. This is an indeterminate form we're going to use L'Hopital's rule. But to apply L'Hopital's rule, we first have to convert this product indeterminate form into a quotient indeterminate form. Recognizing that when it comes to L'Hopital's rule, we're going to have to take the derivative of the numerator and the denominator. So who do we push into the denominator? Do we push the natural log or do we push the x? Well, we have sort of two options. If we push the x into the denominator, we'd end up with a natural log of x over one over x. On the other hand, if we push the natural log into the denominator, we're going to end up with x over one over the natural log of x. And so I really think that the simpler derivative here is going to be if we'd use the first form. The natural log, its derivative is one over x and one over x, since it's a power function, we can calculate its derivative using the power rule. So the first form is going to be a little bit easier. We prefer that one. So applying L'Hopital's rule, because I should mention this thing now has the form as x goes to zero, the numerator will look like negative infinity, the denominator will look like positive infinity. So this has an indeterminate form, infinity over infinity. So we're going to take the derivative of the natural log of x. This will sit above the derivative of x to the negative one, as x approaches zero from the right. And like we mentioned before, the derivative of the natural log is going to be one over x. The derivative of x to the negative one by the power rule is negative x to the negative two, as x approaches zero from the right. Simplifying this thing a little bit, notice, of course, if you have x to the negative two, that's the same thing as having a negative one over x squared. We have a one over x, and then we take the limit as x approaches zero from the right. We have fractions inside of fractions. The easiest way to clean this up is to get times the top and bottom of the big fraction by x squared. This then gives us the limit as x approaches zero from the right. In the numerator, we're just going to get an x. In the denominator, we're going to get a negative one. So as we send x to zero in this situation, we'll get zero over negative one, which is equal to zero, which then turns out to be the true limit. So the takeaway from this example here is that if you have a product indeterminate form like zero over zero times infinity, we can convert that product into a quotient, which will then have the form zero over zero or infinity over infinity. In which case, then we can apply L'Hopital's rule.