Mistakes regarding limits

Concerning concept and calculations

• The most obvious mistake is to think that to calculate the limit of a function is the same thing as to calculate the value of a function in that point. It's not! The function might even be undefined for that point. I have a textbook that explains with this "the limit of a function is a certain value that the function would have to have to be defined there and continuous".

• Another confusion is between the limit and the two sided limits. The two sided limits can exist whereas the limit itself doesn't. If we calculate it from the right and it's equal to [math]\displaystyle{ a }[/math], whereas calculating from the left it's equal to [math]\displaystyle{ b }[/math]. [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] are not the same, hence the limit doesn't exist at that point.

• When a limit results in infinity it's not the same thing as to say that the limit doesn't exist! The fact that the limit of a function goes to infinity means that that limit exits and it's larger than any real number. Or lower in case the limit is negative infinity. When a limit doesn't exist it means that there is a contradiction. Depending on which route we take to approach a certain value the function can assume more than one value, meaning that we can't know for sure if we are reaching a large number, small number or anything in between.
  
  A related conceptual mistake is to think that infinity is an extreme value. No! If the function never stops growing or decreasing, there is no global maximum or global minimum value for it. There may be a local point where a local maximum or local minimum exist though.

• When we learn improper integration there is a similar situation in which we are evaluating an integral when one or both the boundaries are infinity. We can't use the fundamental theorem of calculus and calculate [math]\displaystyle{ F(\infty) }[/math]. However, what we can do is to resort to limits such as to calculate [math]\displaystyle{ \lim\limits_{x \ \rightarrow \ \infty} F(x) }[/math] instead. The difference between calculating the function at infinity and calculating the limit as [math]\displaystyle{ x \ \to \ \infty }[/math] is very subtle!

• With integrals we can write [math]\displaystyle{ \int_{-\infty}^{+\infty} f(x) \ dx }[/math]. However, [math]\displaystyle{ \lim\limits_{-\infty \ \to \ +\infty} f(x) }[/math] is meaningless because there is no such thing as calculating a limit over an interval. Errors like this mean that people don't understand what they are reading.

• [math]\displaystyle{ 0.9999... = 1 }[/math]. There is a common misconception that is to think that this number is not equal to 1 when, in fact, it is equal to 1. First, the sequence of digits is infinite. Second, to think that there exists a last digit of this sequence is wrong. The three dots indicate that there isn't "the last number". When we have something that is infinite, it means that there is no end, no boundary, no number to represent it.

• [math]\displaystyle{ \lim_{x \ \to \ a} f(x) + g(x) = L }[/math]. We cannot do this [math]\displaystyle{ \lim_{x \ \to \ a} f(x) = L - g(x) }[/math] because the limit of the sum is the sum of the limits. This can happen due to the absence of parenthesis. If we do that, conceptually we are making this assumption: [math]\displaystyle{ f(x + 2) = f(x) + 2 }[/math]. This mistake is mostly caused by coincidences, because sometimes wild operations yield the expected result.

• Conjugate technique. One of the most challenging concepts of limits happens when we have a division by zero. Why does the limit exist or why can we do operations such as to multiply by a conjugate to eliminate the division by zero? There isn't any black magic in that. The function is not defined for that point where the division by zero occurs. What we do when we calculate a limit is to consider numbers that are extremely close to that point, which is what allows us to get rid of that division by zero because it never occurs in the first place.

Concerning [math]\displaystyle{ \infty }[/math] itself

• We cannot calculate [math]\displaystyle{ f(\infty) }[/math] for any function because infinity is not a real number. Nor a complex number. It's not even a number to begin with. That's why it's wrong to say that [math]\displaystyle{ f(\infty) = n }[/math] or [math]\displaystyle{ f(x) = \infty }[/math]. A function can map numbers to numbers and infinity can't be mapped like that. What we can do, however, is to calculate limits.

• By definition, infinity it's greater than any real number. If a sum, limit or integral goes to infinity it means that there is no upper boundary to the value. The sum, limit or integral can be as large as we can make it. If we have a negative infinity it means that there is no lower boundary.

• If a sum goes to infinity, or requires infinitely many operations, we can't compute it in finite time. No matter how fast we can calculate it, it'll require an infinite amount of time to be finished. In here we are making the assumption that each calculation takes some time greater than zero. It may be a small fraction, but it's impossible to calculate anything at infinite speed.

• Now comes a natural question. If we compare the set of real numbers and the set of integers, is one larger than the other? That is, are there more elements in one than in the other? There is a way to prove that there are more real numbers than integers, but that proof isn't required in a calculus course. Both sets have an infinite number of elements, how can infinity compare to infinity? In calculus we don't learn how to treat that.

• [math]\displaystyle{ \infty + 1 \gt \infty }[/math] True or false? It may seem true because adding one always make a number larger. However, infinity is not a number! So what does it mean to add one to infinity? There are some exercises in calculus where this happens and as a general rule, infinity plus one doesn't change the fact that infinity is still larger than any real number. Infinity minus one, minus one million, minus any large number, remains a quantity which we cannot write with a finite sequence of digits. That's why intervals that end at infinity cannot end with a closed bracket, it must be open. Else it would mean that we are treating infinity as a number that can be reached.

• There is no definition in calculus to expressions such as [math]\displaystyle{ 1^{\infty} }[/math] or [math]\displaystyle{ \frac{1}{\infty} }[/math]. Infinity times itself or any positive or negative number remains infinity. What we can do with infinity is to calculate infinite sums or limits. Another operation that we can use is to multiply infinity by -1 to change it from positive to negative.

• One hard concept to grasp regarding infinity is that an infinite sum of terms can result in a finite quantity. It's often counter-intuitive because as we subtract infinitely many terms we intuitively expect the result to be zero. Or, if the continue to add infinitely many terms, the result should be infinity. In Calculus, as long as one is able to understand limits, the calculations shouldn't be hard to grasp.

• Subtracting infinity from infinity does not yield zero. Nor multiplying it with zero should yield zero. If we make the assumption that infinity is a very large number it seems natural that any number minus itself should result in zero. Since infinity is not a number we can't do that. Suppose that we have two different quantities, both are infinitely large, but one is larger than the other. What's the difference between them? It can be anything from the smallest possible number to infinity itself!

• One analogy to think about infinity. Think about a super hero universe which happens to have more than one universe. It's a multiverse. We don't know the size of the universe nor whether it is infinite or not. If it's infinite it's natural to think that there exists infinitely many planets, stars and galaxies in it. Now suppose that the universe is finite but the limit of it is located very far away at, say, 1 trillion light years from yourself. At that distance, what does 1 light year look like? Plus one, minus one, it would be insignificant compared to where you are now. Now in our superhero's multiverse each universe can have its own particular size. How can you compare the size of each universe? If they are all infinitely large, how can you know which one is larger?