Perhaps the easiest way to understand limits is to look at what happens if one divides by zero. What is the result for dividing 1 by 0 on a standard calculator? The answer is that it gives an error. Dividing by zero is not an allowed math operation. So no one knows what 1/0 equals, right? Well, sort of. Here is where limits come in handy. What if I calculate the value of 1/x and don't actually make x equal to zero but make it very small? I see that 1/1 = 1; 1/0.1 = 10; 1/0.01 = 100; 1/0.001 = 1,000; 1/0.0001 = 10,000; etc. Although I can never make x equal to zero, I can figure out that the smaller x becomes, the larger result I will get. If I could divide by zero, the smallest possible number, I would get the largest possible number, which is infinitely large. Thus, although I can't divide by zero, I know what the result would be if I could because I know where my answer is headed for progressively smaller values of x: infinity.
At the other end of the scale, I could ask myself what 1/x would be if x were equal to infinity. Again, I can't actually put infinity into the calculator and get an answer. However, if I calculate the results for increasing values of x, I discover that the answer becomes progressively smaller. If I could take x all the way to infinity, I would end up with a result of zero. 1/0 = infinity. 1/infinity = 0. Both those answers are determined by letting x come as close as possible to either zero or infinity and watching where the result is heading. The result of dividing 1 by any number between 0 and infinity will fall somewhere between infinity and zero. Thus, 1/x+1 will approach 1 as x approaches infinity and will approach infinity as x approaches zero. For every positive value of x, 1/x+1 will lie somewhere between 1 and infinity.
I find this process of determining the boundaries of an equation's result useful for answering certain philosophical questions. For example, in my last post I pondered the question of whether it is possible for one person to make another person happy. To find my answer, I assigned the person responsible for creating happiness in the life of another person to provide the maximum service possible to that other person. I made that responsible person into a slave. Attentiveness from this person approaches the maximum humanly possible. I then mentally assessed the happiness of the person receiving the full attention of that slave. My conclusion was that humans have no lack of ability to be dissatisfied even in the face of total attentiveness by the person they have made responsible for their happiness. A slave can't read the mind of the master. A slave can only work so fast and so hard and can be only one place at a time. A slave requires food and sleep. My conclusion was that, because of these limitations and others, an unhappy person will continue to be unhappy even with a fully dedicated slave. Therefore, if even dedicated slave-level service doesn't satisfy an unhappy person, I can conclude that no lesser level of service and attentiveness will make an unhappy person happy. There needs to be another source for happiness. Can a person affect the happiness of another person? Yes. But the other person will always have the capability of being completely unhappy regardless of the level of service rendered.
This is the value of calculus to me. It helps me solve certain mental puzzles by giving me endpoints to the possible range of answers as the unknowns move across their full spectrum of possible values.
x approaches infinity questions:
- What if everybody did it?
- What if I did it perfectly?
- What is the worst that could happen (maximum unpleasantness)?
x approaches zero questions:
- What if I did nothing?
- What if nobody did it?
- What is the best that could happen (minimal unpleasantness)?
After I find the limits, it is then easier to figure out the range of answers for likely values of the unknowns in life's equations, e.g. the level of friendship and service I have to offer to an unhappy person in my life will never be sufficient to make that person happy.
Three semesters of calculus for that. Or at least I haven't noticed any other residual effect of those classes. Does anyone have any other practical uses for higher math?