 Let's recap the major ideas behind section 2.8 in active calculus on using derivatives to evaluate limits. Now this seems a little circular because we defined the derivative as a limit. So how is it we are able to compute limits with derivatives? Well, it's through a tool called L'Hopital's Rule. This rule says that if f and g are differentiable functions at the point x equals a, and if f of a and g of a are both equal to zero, but g prime of a is not zero. That's a lot of conditions that have to be met. But if we meet them all, then the limit as x approaches a of f over g is equal to f prime of a over g prime of a. Also, we note that if f prime and g prime happen to be both continuous at x equals a, which is a pretty mild assumption to make, then the limit as x approaches a of f over g is equal to the limit as x approaches a of f prime over g prime. Now what this is saying is that if you're taking a limit of an expression that involves a quotient of two functions and direct evaluation of the limit gives zero over zero, then we can instead calculate the limit as x goes to a of f prime over g prime. Why is this a big deal? Well, the two derivatives here are often simpler than their parent functions, and sometimes the limit might be very easily computable. A couple of things to note about L'Hopital's Rule. First of all, just because a limit of a fraction comes out to be zero over zero, it does not mean that the limit fails to exist, and it does not mean that the limit equals zero because of the zeros involved, nor does it mean that the limit equals one because you have a thing divided by itself. For example, the limit as x approaches one of two x minus two over x minus one comes out to be zero over zero if you directly evaluate x equals one, but it's easy to see that this limit not only exists, but it equals neither zero nor one. So limits of the form zero over zero are called indeterminate forms because the fact that they come out to zero over zero determines nothing about their value. Such limits might exist, they might not exist, if they do they might equal some number other than zero or one. But L'Hopital's Rule gives us a tool for making the limit of an indeterminate form simpler to evaluate. Second thing to note, L'Hopital's Rule is not the quotient rule. In the rule, L'Hopital's Rule, we are not taking the derivative of the original quotient, whose limit we're interested in. L'Hopital's Rule instead says that the limit of this quotient has the same value as the limit of the quotient of derivatives, and this is all that it says. Again, we are not invoking the quotient rule here, we're using the ratio of two derivative values. A version of L'Hopital's Rule also works if the limit comes out to be quote infinity over infinity, which is another kind of indeterminate form. We'll just state the rule here in that case without any further explanation. As a side item, which turns out to be very important, is this notion of the limit as x approaches infinity. We say that a function f of x has a limit, say L as x approaches infinity, if the values of f can be made as close to L as we want by making x sufficiently large. In late person's terms, this means that as x gets larger and larger, f approaches a stable value of L. A version of this idea is in place also for x approaching minus infinity.