Problem 17.1.1.
Use your favorite computational tool to find a value of \(x\) near \(1\) such that \(\abs{f(x)}\lt10^{-3}\text{.}\)
Section 17.1 A Non-Intuitive Limit
. . . in becoming rigorous, mathematical science takes a character so artificial as to strike everyone; it forgets its historical origins; we see how the questions can be answered, we no longer see how and why they are put.―Henri Poincarè(1854–1912)1
https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/
We began our treatment of limits in Chapter 12 informally because it can take time to develop a mindset appropriate to a thorough understanding of limits. However, by proving theorems using properties of limits which we had not yet shown to be true we’ve left a logical hole in the proofs of nearly every theorem, lemma, and corollary we’ve stated. Bishop Berkeley would be most displeased.
It is time to fill those holes.
Loosely speaking, we know that if, as \(x\) gets “closer and closer” to some real number \(a\text{,}\) the function \(f(x)\) gets “closer and closer” to \(A\) then \(\tlimit{x}{a}{f(x)}=A\text{.}\) This phrase “closer and closer” is the source of the logical holes we need to fill.
To illustrate what can go wrong with the intuitive approach to limits that we’ve used so far, consider the limit \(\limit{x}{1}{f(x)}\text{,}\) when
\begin{equation*}
f(x)
=\frac1\pi\inverse\tan(10^8(x-1)).
\end{equation*}
To get a sense of what this function looks like when we let \(x\) get “closer and closer” to \(1\) we’ve tabulated a few values of \(f(x)\) for \(x\) near \(1\) in the table below.
| \(x\) | \(f(x)\) |
| \(1.5\) | \(4.999\) |
| \(1.4\) | \(4.999\) |
| \(1.3\) | \(4.999\) |
| \(1.2\) | \(4.999\) |
| \(1.1\) | \(4.999\) |
| \(1.01\) | \(4.999\) |
| \(1.001\) | \(4.999\) |
| \(1.0001\) | \(4.999\) |
Seems pretty convincing doesn’t it? Can we conclude from this table that
\begin{equation*}
\limit{x}{1}{\frac1\pi\inverse\tan(10^8(x-1))} = 4.999?
\end{equation*}
(Or maybe, that it is equal to 5?)
Sadly, no. In fact, since \(f(x)\) is continuous this limit is equal to zero because
\begin{equation*}
f(1) =
\frac1\pi\inverse\tan(10^8(1-1))% =\frac1\pi\inverse\tan(10^8(0))
=\frac1\pi\inverse\tan(0)=0.
\end{equation*}
The problem with our example is that none of the \(x\) values in the first column is close enough to \(1\text{.}\) Sure, the numbers \(1\) and \(1.0001\) are very close together. But to evaluate this limit we don’t just want to get close, we want to get close enough. For this particular function we’d have to get much closer to \(1\) before we start to see the values of \(f(x)\) getting close to \(0\text{.}\)
And that’s the problem. The nature of the function we’re taking the limit of must be taken into account when we decide what “close enough” means for any particular limit. This isn’t as bad as it sounds, but as always precision is crucial. We need a definition of limit that doesn’t depend on the nature of the function we’re investigating. A useful definition will also recover the Differentiation Rules in a manner that even Bishop Berkeley would agree is valid.
We will begin with limits “at infinity” because, paradoxically, these are often the easiest to understand. As we proceed through examples the question you want to keep in the back of your mind is, “For this problem how close is close enough?”
