What is the limitation of the concept of limit in mathematics?

Answer 1

I had been through the concept of limit in the mathematics. It looks that it is more so depends on the psychology than the logic. Human brain can not handle the very small number and very large number. That means human brain is incapable of handling very small and very large numbers. Very small numbers are accepted as zero and very large numbers are accepted as infinite by the human brain. There are three rules in mathematics. If you divide any number by same number, you get one. You divide zero by any number, you get zero. You divide any number by zero, you get infinity. So when you divide zero by zero, you get any number from zero to infinite number, as per the above rules. The mathematicians are looks to be emotionally involved in finding the answers by the way of the method called as limit. Their brain can not handle the process of limit. So they presume the limit. Here I will like to highlight the limitation of the concept of limit. Here you are going from one to zero. In each step you tend to go to the number zero. There are two ways of travelling to zero. By way of arithmetic progression and geometric progression. Suppose you go by arithmetic progression, then in given number of steps you will reach to zero. for example if you travel half the distance, in two steps only you reach the zero. Suppose you take one tenth step each time you reach zero in only ten steps. So the arithmetic progression is not intended in the concept of the limit. It is the geometrical progression, that is intended here. So suppose every time you travel half the distance. So from one you reach the 1/2 in the first step. In the next step you go to 1/4. In the third step you go to 1/8. In the forth step you go to 1/16. In next step you go to 1/32. You divide this number by 2, you get 1/64. Then 1/128. Then 1/256. Then 1/512. Then 1/1024. (Instead of 1/2 you can take 3/4 every time. You may take 9/10 every time. May be 49/50 every time.) You can take such million, billion or trillion steps. Try doing trillion steps. You get tired. Then you say that it becomes zero. Instead of taking trillion steps you take 100000000000000000000 00000000 0000000000000 00000000000 000000000000000000 000000000 0000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000, so many steps. (To complicate the figure, You can take different figures from 0 to 9.) Still you do not reach zero. Here you have put the number 1 and kept the button of zero for a while. My brain can not handle this number. Possibly nobody can handle this number. I can keep on pressing the zero button from morning to evening. I will get very large number. I may keep it pressed from morning to evening for my life. I will get very very larger number. This job can be done for next human generations for next 5,000,000 years. Then you will get too a large number. Still this number in not the infinite number. You can divide the number 1 by 2 that many times. But you will not reach to zero. Definitely you are very close to zero, but not zero. It's too near, yet it's too far. Conversely, you can go to the figure 1 from 0. In the first step you will go to 1/2. In the next step you will go to 3/4. Here you have to add 1/4 in 1/2. In the next step you will have to add 1/8. Then you have to add 1/16. Then add 1/32. Then add 1/64. Then add 1/128. Then add 1/256. Then add 1/512. Then add 1/1024. This process continues. You take any number of steps, but you do not reach the number '1'. 'Practically' or by way of calculations as per 'physics', you reach to the number '1', but not 'mathematically' or 'actually'. You 'tend' to reach there, but you do not 'reach' there. Again, its so near, yet so far, a distance to travel. Your answer can not be 1 at any given step. Practically your answer is correct. But not mathematically or logically. Most people will call it logical. But it's not 'actually' correct. (It is your psychology to dismiss very small number as zero. It is your psychology to dismiss any very large number as infinite. That is the problem with human brain.) Instead of half you can travel 9/10th, 99/100th, 999/1000th or 9999/10000th distance in each step. Still you can not state that after so many steps you reach to zero from number one or vise verse. You will get 'very small' number near to zero or a number close to 1. You can always revert back from that number to original number. You have to multiply that very small number by 2 as many times. If you multiply the original number 1 by 2, repeatedly. You go on getting larger and lager number. But not the 'infinite' number. You multiply zero by any number, you will get zero only. Take the largest number you can take. The answer will be zero. This is the reason that for different calculations in the examples of limit, you get different answers from zero to very large number. Human brain works within it's limitations. So the concept of limit can be applied to 'physics', but not to 'mathematics'. The brain of the psychologist is also a human brain. So if psychologists can not process this information, the fault can be called as 'limitation' of the human brain. It is more related to human psychology, than related to the actual mathematical fact. So the concept of limit in mathematics is questionable.

Answer 2

The concept of a limit works well with a continuous function.

A limit will not work for a function which is undefined at the point of the limit, such as 1/x as x tends to 0. Then, if x tends to 0 from the left, the limit is negative infinity whereas if x approaches from the right, the limit is positive infinity. But f(0) is not defined.

Also, if the function is defined but discontinuous at that point, you will need to consider two separate limits. For example, consider g(x) = 0 for x in [0, 1) and g(x) = 1 for x in [1, 2) - a step function. Then the limit of g(x) as x approaches 1 from the left is 0 which is not g(1) whereas the limit from the right is 1 and equals g(1).