Business Mathematics and Basic Statistics · Ch 10 — Limits and Derivatives

The Idea of a Limit and Limit Notation

When we study how a function $f(x)$ behaves as $x$ gets closer and closer to some fixed value $a$ — without necessarily ever reaching $a$ itself — we are studying the limit of $f(x)$ as $x$ approaches $a$ . The formal notation for this is

\lim_{x \to a} f(x) = L

read as "the limit of $f(x)$ as $x$ tends to $a$ equals $L$ ." This statement says: as we choose values of $x$ closer and closer to $a$ (from either side), the corresponding values of $f(x)$ get closer and closer to the single number $L$ .

Note

A Limit Is About the Approach, Not the Destination

$\lim_{x\to a} f(x)$ can exist and equal $L$ even if $f(a)$ itself is undefined, or even if $f(a) \neq L$ . The limit only asks what value $f(x)$ is heading toward as $x$ closes in on $a$ — not what happens exactly at $a$ . This distinction matters throughout the chapter: several of the standard formulae below (§4, §5) are built exactly to handle expressions that are undefined at the point of interest.

For a business-mathematics student, limits are the gateway idea behind the derivative (§6 onward) — every rate of change (marginal cost, marginal revenue, the speed at which a quantity is changing) is defined using a limit. This section only builds the vocabulary and notation; the working techniques for actually evaluating a limit come in §§3–5.

A limit is said to exist at $x=a$ only when the function approaches the same value $L$ regardless of which side of $a$ we approach from — this two-sided requirement is examined precisely in the next section.

Definition 1Limit of a function

$\lim_{x\to a} f(x) = L$ means the value of $f(x)$ gets arbitrarily close to the single number $L$ as $x$ gets arbitrarily close to $a$ (from either side), whether or not $f(a)$ itself is defined.

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