Perhaps you have pondered the mysteries and strange occurrences surrounding ∞, 1, and 0. These three numbers have baffled mathematicians for centuries, and are the result of much controversy and misunderstanding. And yet, all three of these are essential to calculus and, thus, our everyday lives.
Before we begin we must clarify the difference between indeterminate form and an undefined expression. An undefined expression occurs when you have an expression that suffers from what amounts to an identity crisis: there are two different rules in arithmetic that correspond to the single expression and therefore it is impossible to find an answer without violating one rule, and therefore left “undefined”. A common example of an undefined function is n/0. All of these expressions result in confusion over which rule to follow. On the other hand, an expression is considered to be in indeterminate form if the expression we end with can be any number of answers by simply changing one or two variables. These are slightly more complex as they involve the use of limits, and pertain strictly to them. Examples of indeterminate form include 0/0, ∞/∞, 0 * ∞, ∞ − ∞, 00, 1∞ and ∞0.
There are some indeterminate forms that are also undefined expressions, but not all indeterminate forms are also undefined expressions. Take for example 1∞. It would seem intuitive that 1∞ or 1*1*1*1*1*1… (all the way towards infinity) = 1, and you would be right. Simply put, 1∞ equals 1. So why is it considered indeterminate? Because indeterminate form is dealing with limits and not with simple arithmetic like an undefined expression might be, it is indeterminate. 1∞ is not on its own indeterminate, but it is a type of indeterminate form that, when used in calculus, results in a limit that can be any number you decide.
So, the proof. An example of one type of limit that would be considered in 1∞ indeterminate form would be lim x->0+ of (1+x)(ln(g)/x) = lim x->0+ of (1+x)1/x^ln(g) = eln(g) = g. And this will work for any number g. As you can see, because we can plug any number that we want into the place of g we can arrive at any number g and never be able to know which was the original function. Therefore, this function is indeterminate, and the type of indeterminate form it is in corresponds to the 1∞ variety.
Similar proofs exist for all the different types of indeterminate forms that all result in any number. As the limit approaches that point, you can manipulate the function to give you a desired answer. In conclusion, math is confusing and sometimes appears to contradict itself, but I assure you, everything in math is consistent and clear if you try hard enough. When “the people at the big round table”, as Mrs. Izzo says, decide something, they keep the beautiful cycle of math in its proper order.
