Meaning and Intuitive Idea of a Limit

Business decisions are full of questions about tendency rather than an exact value: as the number of units sold gets closer and closer to a factory's full capacity, what does the average cost per unit approach? As a loan's repayment period stretches out further and further, what does the total interest paid tend towards? Calculus answers exactly this kind of question, and the tool it uses is the limit.

Consider the function $f(x) = \dfrac{x^2-4}{x-2}$. Notice that $f(x)$ is not defined at $x=2$, because the denominator becomes zero there. Yet we can still ask a meaningful question: as $x$ takes values closer and closer to $2$ (without actually equalling $2$), what value does $f(x)$ get closer and closer to?

Whichever side we approach $x=2$ from, $f(x)$ keeps getting closer and closer to $4$ — even though $f(2)$ itself is not defined. We say the limit of $f(x)$ as $x$ tends to $2$ is $4$.

Notation. We write this as $\lim_{x \to 2} f(x) = 4$ and, in general, $\lim_{x \to a} f(x) = L$ to mean: as $x$ approaches $a$ — getting arbitrarily close to $a$ without necessarily reaching it — the value $f(x)$ gets arbitrarily close to the fixed number $L$. Here $a$ is the point at which the limit is taken, and $L$ (when it exists) is the value of the limit.

A limit describes the tendency of a function near a point, not necessarily its value at that point — as the table above shows, $f(2)$ can be completely undefined while $\lim_{x \to 2} f(x)$ still exists and equals a perfectly ordinary number. This distinction is exactly what the Continuity section later in this chapter turns into a precise test.

The Odisha CHSE Std-11 Business Mathematics & Statistics syllabus places limits and continuity at the start of the calculus portion of the course, because the differentiation and integration chapters that follow — and, through them, marginal cost, marginal revenue and area-under-a-curve calculations used throughout business mathematics — are all built directly on the idea of a limit introduced here.