One of the standard things that people tell me about Calculus is that they did fine in algebra and geometry and other high school math, but once they got to Calculus, they hit a proverbial wall and found it impossible to continue. I have been wondering where that wall is, and perhaps I have found at least one possibility in the concept and notation of limits. Having finished the section on tangents, I am now learning about limits. I have encountered the concept before, back in 1958, that is, not 48 years ago but in the pages of my 1958-dated algebra textbook. These limits were those of progressions, which were to be summed up. The limits I am learning about now are of functions and their output.

I am only beginning with this, but I can see where this would stop someone in their tracks. Instead of getting a simple yes/no or number answer, this limit process offers a moving target which keeps getting closer to that limit number L without ever really giving a complete solid result you can write into the box. The notation is also new and somewhat confusing, because an X approaching zero from the "right side" is actually positive and gets a plus sign, though it is moving leftward or in what would be a "negative" direction. And vice versa for a limit approaching the positive from the negative. Everything is in motion here, rather than sitting still in nice orderly ranks. I am making the mental adjustment so that I can live with this. I will not hit the wall, though I may find it closer and closer as if I were approaching, well, a calculus limit.

And there's also new notation to learn. I still find the "sigma" notation for sums of progressions, surrounded by the parameters for starting, doing, and ending the progression, confusing and even scary. There's something about that big Greek letter S which says "foreign difficult math." I am not taking any chances with the new notation I'm learning for limits. I have been reverently copying it over and over again onto my note pages, along with Anton's explanation: "Expression (3) is read: "The limit of f(x) as x approaches x₀ from the right equals L₁." But the little arrow that you write below the "lim" abbreviation is pointing rightward, not leftward! The whole matter of those sub-scripts continues to perplex me; I am definitely not used to them yet. Some texts use x₀ as the "original" quantity and x as the "worked-on" quantity, while others use x₁ as the "earlier" quantity and x₂ as the "next" or "worked-on" quantity. This is like the V₀ and V₁ in my physics problems, which I always read as letters rather than numbers, so that I kept thinking of Vo and Vi as peculiar hair-care products (some of you may remember Vo5 shampoo, which is still available) that somehow got mixed up in physics problems. I have got to get used to all these sub- and super-scripts. The only way for me to do that is to write them over and over again, like a scribe learning the sacred letters, so that they will no longer appear strange and embarrassing to me. They must appear on my papers as if I really knew what they meant.

Posted at 3:30 am | link