Why limits? – Limit Infinite

The concept of limits appears counter intuitive to anyone who is introduced to them for the first time. Typically, students are introduced to this concept in high school in precalculus or calculus itself. I believe the reason for discomfort with this concept is that students are used to regular arithmetic involving operations such as addition, subtraction, multiplication, division, exponentiation and taking logarithms and now, they are introduced to this concept which describes the behavior of a function as the independent variable gets close to a certain value.

Eventually, all students get comfortable with the mathematical definition of limits, but why are they defined in the first place?

The simple reason is to assign a value to a function at points where it would be otherwise undefined. For example, in evaluating $\frac{x^2 - 4}{x-2}$ at $x = 2$ , you encounter the undefined operation $\frac{0}{0}$ . But, in many instances, you want the value of functions at such points. For example, for a given function $f$ , $g(x) = \frac{f(x)-f(x_0)}{x-x_0}$ is undefined at $x=x_0$ . But, it is good a have a sensible definition to this quantity $g(x)$ at $x=x_0$ .

Now, what is a sensible assignment of value of a function, say $f$ , at a point, say $x_0$ where it is normally(using the usual arithmetic operations) undefined? Well, we would naturally look at values of the function $f$ at points immediately “surrounding” $x_0$ . If we are convinced that the values of the function are “around” a particular value, say $l$ , for points around $x_0$ , we could sensibly define the function at $x_0$ to be $l$ itself. This is exactly what the mathematical definition of limit captures. In the above, we said imprecise words such as “surrounding” and “around”. If the values of the function are within $\pm 10$ of $l$ at points within $\pm 5$ of $x_0$ , is it good enough? Or, do we need to be within $\pm 0.000001$ of the value of the function when we are within $\pm 5$ of $x_0$ ? The mathematical definition of limit does not give any numbers. When we say limit of $f(x)$ as $x\to x_0$ is $l$ , we mean that the value $f(x)$ gets closer and closer to $l$ as $x$ gets closer and closer to $x_0$ . Naturally, if the value of $f(x)$ gets closer and closer to $l$ as $x$ approaches $x_0$ – that is, in other words, it is guaranteed for $f(x)$ to be within a certain small difference from $l$ in neighborhoods of $x_0$ , it seems reasonable to define the value of $f(x_0)$ to be $l$ itself.

Since by the usual arithmetic $f(x_0)$ is undefined, rather than defining $f(x_0)$ itself to be $l$ , we instead say that the limit of $f(x)$ as $x\to x_0$ is $l$ , or $\lim\limits_{x\to x_0} f(x) = l$ .

As an example, say, we have the following definition of $h_1(x)$ :

$h_1(x) = \begin{cases} 1 & \mbox{ if }x = x_0\pm\frac{1}{2^n}\mbox{ for some }n\in\mathbb{N} \\ 0 & \mbox{ otherwise }\end{cases}$

Clearly, $\lim\limits_{x\to x_0} h_1(x)$ does not exist.

As another example, consider the function $h_2(x)$ :

$h_1(x) = \begin{cases} x-x_0 & \mbox{ if }x = x_0\pm\frac{1}{2^n}\mbox{ for some }n\in\mathbb{N} \\ 0 & \mbox{ otherwise }\end{cases}$

$\lim\limits_{x\to x_0} h_1(x)$ exists and is equal to $0$ .

Observe that in definition of limits we are not confined to continuous functions (in fact, definition of continuity depends on the concept of the limits, not vice versa). The examples above consist of discontinuous functions. In fact, limits can be defined for functions where the independent variable is “discrete”.

Consider the function (sequence) $p:\{\pm \frac{1}{2^n}\big | n\in\mathbb{N}\}\cup\{0\}\to\mathbb{R}$ defined as:

$p(x) = \begin{cases} x & \mbox{ if }x \ne 0 \\ 0 & \mbox{ otherwise }\end{cases}$

Then, $\lim\limits_{x\to 0} p(x) = 0$ .

Secondly, consider the function (sequence) $q:\mathbb{N}\to\mathbb{R}$ defined as $q[n] = \frac{1}{n}$ . Then, $\lim\limits_{n\to \infty} q[n] = 0$ .

We briefly come back to $g(x) = \frac{f(x)-f(x_0)}{x-x_0}$ . For values of $x\ne x_0$ , but in the neighborhood of $x_0$ , this quantity gives the slope of the secants to the graph of $f(x)$ “at” $x=x_0$ . By making $x$ get closer and closer to $x_0$ , we make the secants become more and more like the tangents to the graph of $f(x)$ at $x=x_0$ . Using the justification given above for concept of the limit, we define the slope of tangent of $f(x)$ at $x = x_0$ to be $\lim\limits_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}$ .

We make one final point about how infinity plays a role in the definition of limit. It seems students are often thrown off by limits just because infinity is involved. As we discussed above, limits deal with neighborhoods of a point ( $x=x_0$ in our discussion above) that get arbitrarily small ( $x$ gets arbitrarily close to $x_0$ ). In order to capture the notion of neighborhoods of $x_0$ that become smaller and smaller (infinitesimally small), the mathematical definition of limit uses infinite sequences of numbers where each number is closer to $x_0$ than the previous number.