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

Left-Hand Limit and Right-Hand Limit

Because $x$ can approach $a$ from two directions — from values less than $a$ or from values greater than $a$ — a limit is more precisely built from two one-sided limits:

The left-hand limit (LHL), written $\displaystyle\lim_{x \to a^{-}} f(x)$ , is the value $f(x)$ approaches as $x$ tends to $a$ through values smaller than $a$ .
The right-hand limit (RHL), written $\displaystyle\lim_{x \to a^{+}} f(x)$ , is the value $f(x)$ approaches as $x$ tends to $a$ through values greater than $a$ .

Note

The Existence Rule

$\displaystyle\lim_{x\to a} f(x)$ exists if and only if the LHL and the RHL both exist and are equal:

$\lim_{x\to a^{-}} f(x) = \lim_{x\to a^{+}} f(x) = L \quad\Longrightarrow\quad \lim_{x\to a} f(x) = L$

If the LHL and RHL exist but are different, the two-sided limit does not exist at that point — even though each one-sided limit is perfectly well-defined on its own.

This distinction is especially important for piecewise-defined functions, which are common in real business situations — a tariff or commission structure is often defined by one rule below a threshold quantity and a different rule at or above it. At the threshold itself, the LHL (computed from the "below" rule) and the RHL (computed from the "at-or-above" rule) may or may not agree, and checking this is exactly how we decide whether the overall limit exists at that breakpoint.

Method for a piecewise function at a breakpoint $x=a$ : …

Definition 2Left-hand limit (LHL)

$\lim_{x\to a^-} f(x)$ — the value $f(x)$ approaches as $x$ tends to $a$ through values s …

Definition 3Right-hand limit (RHL)

$\lim_{x\to a^+} f(x)$ — the value $f(x)$ approaches as $x$ tends to $a$ through values greater than $a$ . The two-sided limit exist …

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