Since the beginning of my undergraduate studies in Physics, I have found all kind of doubts related to basic math needed for the effective learning of Differential and Integral Calculus. As many people share an infinity of troubles with this discipline, I decided to write some study notes about critical points in Calculus. My goal here is offering students an intuitive point of view about how to deal with many abstract concepts that are, in practice, complicated to understand at first sight.
I will start exploring one of the most abstract concepts in Calculus, that is known as the formal definition of limit with $\epsilon$ and $\delta$. But instead of starting directly with this definition, I'm going to say something about mathematical functions and how important they are to the learning of Calculus.
Function is one of the most important concepts in math. There are many ways to define it, one of them is to imagine that we have two groups of things. So one of these groups we can describe with $y$ (first group) and the other we will describe with $x$ (second group). The first group is related with the second through a tool, or, in a better way, a function that we will call $f(x)$, what means that when I choose something in the second group the first group will, somehow, respond directly. In a mathematical way, we can express this relation between groups by:
\begin{equation}
f(x) = y.
\end{equation}
Now, if those groups are composed by numbers, if you apply any $x$ to $f(x)$ you will obtain $y$. Therefore, if I apply different values of $x$ in $f(x)$ you will obtain different values of $y$ (it is important to say that a number in the second set of numbers is related with only one number of the first set, and this is a property of the space os functions). In this way, we can represent this equivalence between $x$ and $f(x)$, or better, between $x$ and $y$ with a cartesian plan as we can see in the Figure 1.
As an example, if $x = x_0$, then $f(x) = f(x_0) = y_0$, as seen in Figure 2.
Now we can introduce the idea of limit. Why is it important to calculate the limit of a function for a given $x$? Let's ilustrate with an example. Consider the following function:
\begin{equation}
f(x) = \frac{1}{x},
\end{equation}
if:
\begin{equation}
x = 1, f(1) = 1, \\
x = 2, f(2) = \frac{1}{2}, \\
x = 3, f(3) = \frac{1}{3}, \\
.\\
.\\
.\\
x = n, f(n) = \frac{1}{n}.
\end{equation}
Until here, alright, ok? I mean, if $x$ is a really big number (I'm saying that $x \rightarrow \infty$), then $f(x)$ will be a very small number (I'm sayng that $f(x) \rightarrow 0$). However, what happens with $f(x)$ if $x$ becomes very small? Let's see:
\begin{equation}
x = 0.9, f(0.9) = 1.111, \\
x = 0.8, f(0.8) =1.25, \\
x = 0.5, f(0.5) = 2, \\
x = 0.005, f(0.005) = 200, \\
x = 0.00001, f(0.00001) = 100000, \\
\end{equation}
So, when $x$ becomes very small ($x \rightarrow 0$), $f(x)$ becomes very big ($f(x) \rightarrow \infty$)! It can be seen by the Figure 3.
As we can notice, the point $x = 0$ has no analytical value, and it is called singularity. When it happens, is said that the function $f(x)$ is not analytical in $x = 0$ because it diverges! Then, I ask you the following question: what is the value of $f(x)$ if $x \rightarrow 0$? Or, in another way,
\begin{equation}
\lim_{x \rightarrow 0} \frac{1}{x} = ?
\end{equation}
Trying to solve this question (or questions correlated with this one), mathematicians and physicists developed the Calculus. Historically, the development of the Calculus started with Isaac Newton when he wanted to describe the behavior of the nature from the smallest particles to motion of the planets. There are many controversies about who started the development of the Calculus, if was Newton or Leibniz (here and here you can read more about this complicated panorama).
Anyway, keeping this ideas in mind, I will explore another abstract concept about Set Theory for, after, entering at the formal definition of limit. We usually deal with the following interesting sets of numbers:
Naturals: $(0,1,2,3,..., \infty)$,
Integers: $ (-\infty,..., -3, -2, -1, 0, 1, 2, 3, ..., \infty)$,
Rationals: all the integers and finite decimals,
Irrationals: infinite decimals and non-periodic numbers like $\pi = 3.1415...$,
Reals: all those sets are here.
The most general way to characterize a number line is using the real set of numbers (Figure 4). The impressive thing about the reals is that there are infinite points between two points!!! And This is amazing! It means that in a numerical range it is always possible to find a small value, as small as you want! And from here we can start with the formal definition of limit.
The formal definition of limit says that, for the existence of $\lim_{x \rightarrow a} = L$, the following must be truth: for all $\epsilon > 0$ must be $\delta > 0$ in a way that the range $|x-a|$ need to be less than $\delta$ and the range $|f(x) - L|$ need to be less than $\epsilon$. What can be expressed (or written) mathematically by:
\begin{equation}
\exists \lim_{x \rightarrow a} = L \Leftrightarrow \forall \epsilon > 0 \exists \delta > 0 ; |x-a|< \delta \Rightarrow |f(x)-L|< \epsilon.
\label{limdef}
\end{equation}
It is important to notice that $\epsilon$ and $\delta$ are arbitrary numbers (as small as you want) , which means that you can choose them according to your needs, but they must satisfy the limit definition. Furthermore, if I want to calculate the limit of a strange function that has discontinuity in a given point, so I can try to figure out the value of this function in near ranges of the point, as close as I want ($\delta$) and the result of $f(x)$ will be as close of the real value as I want ($\epsilon$).
We can see how limit definition works on a practical example! Let's prove with the limit definition the following limit:
\begin{equation}
\lim_{x \rightarrow 1} 2x + 1 = 3.
\end{equation}
Sol.:
Here we have $a = 1$, $f(x) = 2x+1$ and $L = 3$. The limit definiton is expressed by \ref{limdef}. Then, let's check the ranges:
\begin{equation}
|x-a| = |x-1| < \delta,
\label{delta}
\end{equation}
\begin{equation}
|f(x) - L| = |2x+1 - 3|< \epsilon.
\label{epsilon}
\end{equation}
We can manipulate \ref{epsilon} to see if it can appears like \ref{delta}.
\begin{equation}
|2x-2| < \epsilon \\
|2(x-1)|< \epsilon \\
|x-1| < \frac{\epsilon}{2}.
\end{equation}
Here! There is a range in $y$ axis such that $|f(x) - L|$ is less than $\epsilon$ and this value is $\frac{\epsilon}{2}$. It can be seen in Figure 6.
Now, I need to find a value for $\delta$ that satisfies a range in $x$ axis. For this, we can notice that if I choose $\delta = \frac{\epsilon}{2}$, then the range in the $x$ axis will be satisfied, implying that:
\begin{equation}
|x-1| < \delta = \frac{\epsilon}{2}.
\end{equation}
Therefore, if $\delta = \frac{\epsilon}{2}$, so I can choose a value for $\epsilon$ as small as I want, for exemple, $\epsilon = 0.00000001$, in a way that the limit definition is satisfied!
Now, I leave here a problem for you:
Prove, by the limit definition, that:
\begin{equation}
\lim_{x \rightarrow x_0} \sum_{i=0}^{n} a_i x^i = \sum_{i=0}^{n} a_i x_{0}^{i}.
\end{equation}
And comment below if you want to discuss!
See you!
PS: I would like to thank Amanda for the drawings and the great writing review.
