One-Sided & Infinite Limits — One-Sided Limits and Infinite Limits - Business Calculus

SECTION 1

Historical Context & Motivation

The notion of a limit lies at the very foundation of calculus, yet its precise formulation took centuries of intellectual struggle. Ancient Greek mathematicians such as Archimedes used a method of exhaustion that anticipated limits by trapping areas between inscribed and circumscribed polygons, but they lacked a formal language for what happens when a quantity approaches, rather than reaches, a target. It was not until the seventeenth century that Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus, relying on intuitive notions of 'infinitely small' quantities—infinitesimals—that provoked philosophical criticism for their logical gaps. The rigorous ε–δ definition of limits eventually supplied by Augustin-Louis Cauchy and Karl Weierstrass in the nineteenth century resolved these concerns and opened the door to careful treatment of one-sided limits and infinite limits—situations where a function's behavior differs on each side of a point, or where the function grows without bound.

~250 BCE • Archimedes' Method of Exhaustion

Archimedes approximated areas and volumes using polygonal bounds, foreshadowing the concept of a limit without ever naming it.

1684–1687 • Newton & Leibniz Invent Calculus

Both mathematicians introduced derivatives and integrals using loosely defined infinitesimals, generating powerful tools but also logical controversy.

1821 • Cauchy's Cours d'Analyse

Cauchy published a rigorous definition of the limit and began formalizing one-sided limits, laying the groundwork for modern analysis.

1860s • Weierstrass Completes the ε–δ Framework

Weierstrass refined the epsilon-delta definition, enabling precise discussion of infinite limits, one-sided limits, and continuity.

Why do one-sided and infinite limits matter in a business calculus course? Many real-world business models contain breakpoints—tax brackets that change at specific income levels, shipping cost functions that jump at weight thresholds, or profit curves that exhibit vertical asymptotes near capacity constraints. A two-sided limit alone cannot capture what happens at these critical junctures. By introducing left-hand and right-hand limits, we gain the ability to analyze each direction of approach independently. And by defining infinite limits, we can describe scenarios in which costs, revenues, or other quantities grow without bound as a variable nears a particular value—an essential concept for identifying vertical asymptotes on graphs and understanding the boundaries of feasible models.

SECTION 2

Core Principles & Definitions

Before diving into formal notation, it helps to anchor three fundamental ideas: limits describe approach, not arrival; direction matters; and unbounded growth is a valid limiting behavior. The cards below summarize the foundational concepts upon which the rest of this lesson is built.

Left-Hand Limit

The value f(x) approaches as x → a from values less than a. Written lim (x→a⁻) f(x). Only x-values to the left of a are considered.

Right-Hand Limit

The value f(x) approaches as x → a from values greater than a. Written lim (x→a⁺) f(x). Only x-values to the right of a are considered.

Two-Sided Limit Existence

The ordinary (two-sided) limit lim (x→a) f(x) = L exists if and only if both one-sided limits exist and are equal: lim (x→a⁻) f(x) = lim (x→a⁺) f(x) = L.

Infinite Limit

We write lim (x→a) f(x) = +∞ (or −∞) when f(x) increases (or decreases) without bound as x approaches a. The limit does not exist as a real number, but the notation conveys the function's unbounded behavior.

Vertical Asymptote

The vertical line x = a is a vertical asymptote of f if at least one of the one-sided limits at a is +∞ or −∞. The graph of f approaches but never crosses this line near x = a.

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KEY TAKEAWAY

Think of one-sided limits like approaching a traffic intersection from two different streets. Even though both streets lead to the same intersection (the point x = a), traffic from the east may be flowing smoothly at 30 mph while traffic from the west is backed up and barely moving. If the 'speeds' from both sides don't match, the overall traffic condition at the intersection is undefined—just as the two-sided limit fails to exist when the left-hand and right-hand limits disagree.

SECTION 3

Visual Explanation — One-Sided Limits

The diagram below illustrates a piecewise function with a jump discontinuity at x = 2. Notice how the left-hand branch and the right-hand branch approach different y-values at the same x-coordinate. The open circle marks a value the function does not attain from that direction, while the filled circle marks the actual function value (or the value approached). This visual distinction is central to understanding why the two-sided limit does not exist at x = 2, even though each one-sided limit is perfectly well defined.

The cyan curve shows f(x) for x < 2, approaching y = 2 (open circle). The pink curve shows f(x) for x > 2, approaching y = 4 (filled circle). Because 2 ≠ 4, the two-sided limit at x = 2 does not exist.

In the diagram above, the dashed violet vertical line at x = 2 marks the point of interest. The open circle on the cyan curve at (2, 2) indicates that the function does not actually equal 2 when x = 2—the left-hand branch merely approaches that value. Meanwhile, the filled pink circle at (2, 4) shows the actual function value f(2) = 4, which coincides with the right-hand limit. This is a characteristic signature of a jump discontinuity: both one-sided limits exist as finite numbers, but they disagree. In business contexts, such jumps appear in step-pricing models, tax brackets, and tiered commission structures where rates change abruptly at defined thresholds.

SECTION 4

Mathematical Framework

We now formalize the intuitive ideas from the previous sections using precise notation. Each definition below is stated in terms of the ε–δ language (or its infinite-limit analogue), which is the standard of rigor in university-level calculus. Understanding these formal statements is not just an academic exercise—it gives you the tools to prove limit values, identify when limits fail to exist, and construct convincing arguments in applied settings.

LEFT-HAND LIMIT

lim (x→a⁻) f(x) = L means: ∀ ε > 0, ∃ δ > 0 such that 0 < a − x < δ ⟹ |f(x) − L| < ε

Here x approaches a only from values less than a. The condition 0 < a − x < δ restricts attention to a left-sided neighborhood of a.

RIGHT-HAND LIMIT

lim (x→a⁺) f(x) = L means: ∀ ε > 0, ∃ δ > 0 such that 0 < x − a < δ ⟹ |f(x) − L| < ε

Now x approaches a only from values greater than a. The condition 0 < x − a < δ restricts attention to a right-sided neighborhood.

INFINITE LIMIT (POSITIVE)

lim (x→a) f(x) = +∞ means: ∀ M > 0, ∃ δ > 0 such that 0 < |x − a| < δ ⟹ f(x) > M

M replaces ε because we no longer seek closeness to a finite number L. Instead, we demand that f(x) exceeds every arbitrarily large bound M whenever x is sufficiently close (but not equal) to a.

INFINITE LIMIT (NEGATIVE)

lim (x→a) f(x) = −∞ means: ∀ M < 0, ∃ δ > 0 such that 0 < |x − a| < δ ⟹ f(x) < M

Analogous to the positive case, but f(x) falls below every arbitrarily large negative bound M. One-sided infinite limits are defined similarly by replacing |x − a| with the appropriate one-sided condition.

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Important Nuance

When we write lim (x→a) f(x) = +∞, the limit does not exist in the strict sense because +∞ is not a real number. The notation is a shorthand that conveys a specific type of divergence. In contrast, a limit that oscillates (like sin(1/x) as x→0) also does not exist, but for an entirely different reason—it fails to settle on any single behavior at all.

SECTION 5

Classifying Limit Behaviors

When investigating limits, several distinct scenarios can arise, and it is helpful to classify them systematically. The diagram below illustrates three common cases: a function with a removable discontinuity (where both one-sided limits agree but the function value differs or is missing), a jump discontinuity (where one-sided limits exist but disagree), and a vertical asymptote (where at least one side produces an infinite limit). Understanding this classification helps you diagnose limit behavior rapidly in both exam and applied settings.

Three panels compare limit behaviors at x = a. Left: removable discontinuity—both sides agree on L but f(a) ≠ L. Center: jump discontinuity—one-sided limits differ. Right: vertical asymptote—function grows without bound.

Behavior	Left-Hand Limit	Right-Hand Limit	Two-Sided Limit
Removable	L (finite)	L (same finite)	Exists = L
Jump	L₁ (finite)	L₂ ≠ L₁ (finite)	Does not exist
Vertical asymptote	±∞ (or finite)	±∞ (or finite)	Does not exist (as real number)
Oscillation	Does not exist	Does not exist	Does not exist

Summary of limit behavior classifications

SECTION 6

Worked Examples

Example 1: One-Sided Limits of a Piecewise Function

A company's cost function (in thousands of dollars) for producing x units is defined piecewise:

PIECEWISE COST FUNCTION

C(x) = { 2x + 1, if x < 3; x² − 2, if x ≥ 3 }

Find lim (x→3⁻) C(x), lim (x→3⁺) C(x), and determine whether lim (x→3) C(x) exists.

One-Sided Limits of C(x) at x = 3

1
Step 1 — Identify the left-hand rule

For x < 3, we use C(x) = 2x + 1. As x approaches 3 from the left, we substitute x = 3 into this expression (since it is a polynomial and hence continuous on its domain).

lim (x→3⁻) C(x) = 2(3) + 1 = 7
2
Step 2 — Identify the right-hand rule

For x ≥ 3, we use C(x) = x² − 2. As x approaches 3 from the right, substitute x = 3 into this polynomial.

lim (x→3⁺) C(x) = (3)² − 2 = 9 − 2 = 7
3
Step 3 — Compare one-sided limits

Since lim (x→3⁻) C(x) = 7 and lim (x→3⁺) C(x) = 7, the one-sided limits are equal.

lim (x→3) C(x) = 7 (the two-sided limit exists)

Example 2: Infinite Limit and Vertical Asymptote

Consider the average cost per unit function A(x) = 500 / (x − 10) for a firm that cannot produce exactly 10 units due to a production constraint. Let us evaluate the one-sided limits at x = 10.

Infinite Limit of A(x) at x = 10

1
Step 1 — Analyze left-hand behavior

When x approaches 10 from the left (x < 10), the denominator (x − 10) is a small negative number. Thus 500 divided by a small negative number yields a very large negative value.

lim (x→10⁻) 500/(x − 10) = −∞
2
Step 2 — Analyze right-hand behavior

When x approaches 10 from the right (x > 10), the denominator (x − 10) is a small positive number. Hence 500 divided by a small positive number yields a very large positive value.

lim (x→10⁺) 500/(x − 10) = +∞
3
Step 3 — Conclusion

Because the left-hand limit is −∞ and the right-hand limit is +∞, the two-sided limit does not exist even in the extended sense. However, since at least one side diverges to ±∞, we conclude that x = 10 is a vertical asymptote of A(x).

x = 10 is a vertical asymptote of A(x)

SECTION 7

Strengths & Limitations of One-Sided and Infinite Limit Analysis

One-sided and infinite limits are powerful diagnostic tools, but like any mathematical technique, they carry both strengths and limitations. The table below provides a concise comparison to help you understand when each concept is most useful and where its boundaries lie.

Aspect	Strengths	Limitations
One-Sided Limits	Detect jump discontinuities; determine two-sided limit existence; model piecewise and step functions common in business pricing, taxation, and insurance.	Cannot detect oscillatory behavior (e.g., sin(1/x) near 0); require knowledge of the function rule on each side; may be tedious for complex piecewise definitions.
Infinite Limits	Identify vertical asymptotes; reveal domain restrictions; useful in cost-per-unit and rate-of-change models where denominators approach zero.	±∞ is not a real number, so standard limit laws (sum, product) do not automatically apply; can lead to indeterminate forms like ∞ − ∞ that require further analysis.
Two-Sided Limits	Simplest to compute for continuous functions; foundation for the derivative definition; widely applicable.	Insufficient alone for piecewise or discontinuous functions; may mask asymmetric behavior that one-sided analysis would reveal.

Comparison of limit analysis approaches

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WHEN TO USE WHICH

In a business calculus setting, think of one-sided limits as the 'audit' you perform whenever a function changes its rule at a threshold—a new tax bracket, a volume discount cutoff, or a regulatory cap. Infinite limits, on the other hand, serve as early-warning signals: they flag parameter values where your model breaks down, such as production levels that drive per-unit costs through the roof or advertising budgets that produce diminishing and ultimately unbounded marginal costs.

SECTION 8

Connection to Continuity, Derivatives, and Limits at Infinity

One-sided and infinite limits are not isolated concepts; they weave directly into the larger tapestry of calculus. Understanding them is a prerequisite for defining continuity at a point, constructing the derivative as a limit of a difference quotient, and later analyzing limits at infinity (horizontal asymptotes). The table below shows how today's concepts extend into more advanced theory.

Concept (This Lesson)	Advanced Extension	Connection
One-sided limits	Continuity from the left/right	f is continuous at a iff lim(x→a⁻) f(x) = lim(x→a⁺) f(x) = f(a). One-sided limits are the building blocks of continuity.
One-sided limits	Left/right derivatives	At endpoints of closed intervals (e.g., [0, T] in a business planning horizon), derivatives are computed as one-sided limits of the difference quotient.
Infinite limits (x→a)	Limits at infinity (x→∞)	Infinite limits identify vertical asymptotes. Limits at infinity identify horizontal asymptotes—the long-run behavior of revenue, cost, or profit functions.
Vertical asymptotes	Improper integrals	Integrating a function with a vertical asymptote inside the interval requires splitting the integral and evaluating one-sided limits—an improper integral.

How one-sided and infinite limits connect to later topics

In the coming lessons, you will see that the definition of the derivative f ′(a) = lim (h→0) [f(a+h) − f(a)]/h is itself a two-sided limit of a difference quotient. If f has a corner or cusp at x = a, the left-hand derivative (h→0⁻) and the right-hand derivative (h→0⁺) will disagree, and the derivative at a will fail to exist—mirroring exactly the logic of one-sided limits you have learned here. Mastering the present topic therefore provides essential practice for the differentiation framework you will encounter next.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain in your own words why a two-sided limit can fail to exist even when both one-sided limits exist and are finite. Provide a real-world business scenario in which this situation could arise.

Show answer

A two-sided limit fails to exist when the left-hand limit and the right-hand limit are both finite but unequal—this is a jump discontinuity. The two-sided limit requires agreement from both directions. For example, consider a shipping cost function that charges $5 per unit for orders under 100 units and $3 per unit for orders of 100 or more. As order size approaches 100, the cost-per-unit from the left approaches $5 while from the right it equals $3. The two-sided limit at x = 100 does not exist because the one-sided values disagree.

Question 2: BASIC CALCULATION

Let g(x) = { 3x − 1 if x < 2; x² + 1 if x ≥ 2 }. Find lim (x→2⁻) g(x), lim (x→2⁺) g(x), and determine whether lim (x→2) g(x) exists.

Show answer

For the left-hand limit, use g(x) = 3x − 1: lim (x→2⁻) (3x − 1) = 3(2) − 1 = 5. For the right-hand limit, use g(x) = x² + 1: lim (x→2⁺) (x² + 1) = (2)² + 1 = 5. Since both one-sided limits equal 5, the two-sided limit exists and lim (x→2) g(x) = 5. Note that g(2) = 5 as well (from the x ≥ 2 rule), so g is actually continuous at x = 2.

Question 3: INTERMEDIATE

Evaluate lim (x→−4⁺) (x + 1) / (x + 4) and lim (x→−4⁻) (x + 1) / (x + 4). Based on your results, determine whether x = −4 is a vertical asymptote.

Show answer

As x → −4⁺, the numerator approaches −4 + 1 = −3 (a negative number), and the denominator (x + 4) approaches 0 through small positive values. Thus the ratio tends to −3 / (0⁺) = −∞. As x → −4⁻, the numerator still approaches −3, but the denominator approaches 0 through small negative values, giving −3 / (0⁻) = +∞. Since at least one one-sided limit is ±∞, x = −4 is indeed a vertical asymptote of f(x) = (x + 1)/(x + 4). In fact, both sides diverge to infinity (in opposite directions).

Question 4: APPLIED

A small firm's profit function (in thousands of dollars) is P(x) = 50x / (200 − x), where x is the number of units produced (0 ≤ x < 200). Interpret the behavior of P as x → 200⁻. What does this suggest about the firm's production capacity?

Show answer

As x → 200⁻, the numerator 50x approaches 50(200) = 10,000 (positive), and the denominator (200 − x) approaches 0 through small positive values. Therefore lim (x→200⁻) P(x) = +∞. In practical terms, the model predicts that profits grow without bound as production approaches 200 units. This signals that x = 200 is a vertical asymptote and represents the firm's theoretical production capacity—a hard constraint beyond which the model is no longer valid. The infinite limit warns us that the model breaks down near this boundary and should be interpreted with caution.

Question 5: CRITICAL THINKING

Construct a single function f(x) defined on all of ℝ \ {1} that satisfies all three conditions simultaneously: (i) lim (x→1⁻) f(x) = 3, (ii) lim (x→1⁺) f(x) = +∞, and (iii) f(0) = 2. Prove that your function meets each condition, and classify the discontinuity at x = 1.

Show answer

One valid construction is f(x) = { 3 if x < 1; 1/(x − 1) + 3 if x > 1 } with f(0) = 3 from the first piece, which conflicts with condition (iii). Adjusting: let f(x) = { 2 + x if x < 1; 1/(x − 1) if x > 1 }. Then f(0) = 2 + 0 = 2, satisfying (iii). For (i): lim (x→1⁻) (2 + x) = 3 ✓. For (ii): as x → 1⁺, 1/(x − 1) → +∞ ✓. The discontinuity at x = 1 is classified as an infinite (essential) discontinuity because the right-hand limit is +∞. It is not removable (the limits don't agree on a finite value) and not a jump (a jump requires both sides finite). This type of discontinuity corresponds to a vertical asymptote at x = 1 from the right side.

SUMMARY

Lesson Summary

Summary

This lesson introduced two essential extensions of the basic limit concept. A left-hand limit (lim x→a⁻) examines function behavior as x approaches a from values less than a, while a right-hand limit (lim x→a⁺) examines behavior from values greater than a. The two-sided limit exists if and only if both one-sided limits are equal. When one-sided limits disagree, we encounter a jump discontinuity—a frequent occurrence in piecewise business models such as tiered pricing, tax brackets, and commission structures.

An infinite limit arises when f(x) grows without bound as x approaches a finite value a, signaling a vertical asymptote on the graph. Determining the sign of each one-sided infinite limit (+∞ or −∞) requires analyzing the signs of the numerator and denominator near x = a. These tools prepare you for the definitions of continuity and the derivative, both of which rest fundamentally on limit analysis.