I have not been able to replicate my early successes in solving limit problems. They all seem to get away from me. What's more, they are consumable. Once I've either solved or failed one, I can't go back and solve it again until I've forgotten it, which will take about a week. Meanwhile the problems in the set, in the unbreakable tradition of math problem sets, get harder and harder as you go on, so that if I missed the early ones I will not have an easier time with the later ones. Thus I must quest for more introductory calculus problems, hoping that sooner or later I will figure out how to do them again.

One of the reasons I'm not doing so well as I did before was that earlier on, I was doing them simply by rote, in a mechanical way, as I learned in Anton's book. I had no idea why I should do the work, only that I should do it. But why do it that way? When do you factor out the algebra, and when do you leave it alone? (Why am I learning calculus? Because of a challenge to myself many years ago. In 2001, a blind man climbed to the summit of Mount Everest. If he could do that, I could learn mathematics and physics.)

I had recourse to almighty Google (Please don't be evil, even if you can buy the whole Internet!) and typed in "Calculus Problems." Up came a site from the University of California at Davis (where, coincidentally, my geo-chemist cousin teaches in the Geology department) which offered typical lists of calculus problems. I addressed the first set of "Limits of Functions as X approaches a Constant" problems and promptly got lost. I needed to read the introduction again. It always helps to read the instructions. Sometimes I forget that. This time the virtual professor offers up this advice.

"…In fact, the form "0/0" is an example of an indeterminate form. This simply means that you have not yet determined an answer. Usually, this indeterminate form can be circumvented by using algebraic manipulation. Such tools as algebraic simplification, factoring, and conjugates can easily be used to circumvent the form "0/0" so that the limit can be calculated."

So that's why you have to factor those polynomials out. Why couldn't the book just tell me that, or did I miss it? And the using isn't so easy, when I haven't done that type of factoring for about four years. I dusted off my beautifully calligraphed algebra notes and found out how I deconstruct this or that. And then after you've done the work, then you crank the Newtonian wheels and out pops the limit. But I've consumed the problem. Now I can't solve it again for a while, and I need to find more fresh calculus problems, rather like a squirrel foraging for acorns in the autumn woods.

Posted at 2:58 am | link