Alright, in this 7th edition of this Uni text, I happened to discover something new to mathematics-- that the derivative as dy/dx is an angle dy divided by a angle dx. Sure, the derivative can be a length dy divided by length dx, but when we want to relate the integral to the derivative as duals of one another (not as inverses but duals) we have to describe the derivative as a concept to the integral's area. The best description of the derivative is that it is angles versus the integral's area. So, I am sorry for the disorder and disorganization of the 7th edition of the Uni text at this moment, because of the new discovery, there is bound to be an amount of disorder, so let me try to gain back the order with the function y = 1/x and the concept of the limit in Old Math as we are still on the derivative.
Now we have the function y = 1/x plotted in the 1st quadrant only in the 10 Grid and we have exactly a total of 100 cells along the x-axis and have used up 10 cells to get to x=1, y=1. So we have 90 cells remaining before we exhaust all the cells in 10 Grid by y = 1/x.
So, we have 90 cells for the function to reach infinity border of 10 and on the y-axis, how many possible points can we use altogether in the 10 Grid? I want the student and reader to stop and think of what possible y value points can be used for y = 1/x starting at x = 1 and ending at x = 10 in those 90 cells remaining?
Alright, after you stopped and thought about it, you would realize that there are just 11 possible numbers for the y value as 0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0, yet, there are 90 cells remaining of 90 unique x values.
So, what does that mean? Well, it means that there are going to be a lot of repeats of y-values for successive x values. For example x=3.0 y = .3, and then x=3.1, y=.3. Remember in 10 Grid, we truncate for 1/100 values do not exist in 10 Grid, but exist in 100 Grid.
So, now, we know that the function y = 1/x looks more smooth as a "curve" if we did not have so many straightlines in the 10 Grid. For example, drawing a circle in the 10 Grid is going to have a "boxy or square look" to it and only when we go to the 100 Grid, which our Computer Screen is based on 100 Grid that we "curve out those straightline segments".
Always keep in mind that in true-calculus, true-math, that curves do not exist at all. Curves our all composed of tiny straightline segments and it is our eyes that deceive us into thinking "that is a curve". If you see a straightline in a circle arc, you just have not the proper Grid system.
So now, we introduce the Superposition concept into Calculus, and we toss out or throw out the limit concept as fake garbage. I like to talk about the limit concept of Old Math in an analogy. In Africa they had voodoo witchdoctors whenever people got sick with say a virus or bacteria or kidney stone or other medical ailment, and the voodoo witchdoctor would come into the scene and perform a dance and ritual for hours and hours. Whereas the real medical doctor or scientist doctor would come on the scene and diagnosis the ailment and try to fix the person via medical science. The real doctor would not be spending time on irrelevant things, like dancing or concocting herbs and other irrelevancies. Well, hate to say it, but in modern mathematics with the limit concept of Calculus is a massive irrelevancy in doing mathematics. Everytime a calculus teacher or text with limit concept goes into action, is no better than if they did a voodoo witchdance in front of the classroom.
What is relevant though, for the function y = 1/x, is to superimpose the next higher Grid, whenever we see too many straightlines. So that we still plotted the function y =1/x in the 10 Grid, but now we use the Superposition concept to impose the 100 Grid in cases when we have two successive cells have the same y value such as x=3.0, y = .3 and then x=3.1, y=.3 and then x=3.2, y =.3.
So, we superimpose the 100 Grid upon the 10 Grid and we have x=3.0, y=.33 and then x=3.1, y =.32 and then x=3.2, y =.31.
Now if we started with the 100 Grid rather than the 10 Grid, we probably would not need to superimpose the 1000 Grid to make the function look more curvy. For in fact, since our Computer screen makes circles look curvy and deceive our eyes by not seeing the tiny straightlines, we can be well assured that the 100 Grid makes the function y = 1/x look curvy.
So, here in New Math, in True Calculus, in True Math, we do not teach young students a fakery of Calculus with a hideous irrelevant limit concept, where you dance around in a voodoo dance a neighborhood of the x axis and a neighborhood of the y axis. In New Math, we teach people, young people science, real science and not the fakery of limit.