Source: Calculus for Business, Chapter 1 (FIU)
Tags: limits, continuity, one-sided limits, two-sided limits, limit laws, continuous function, piecewise continuity, calculus for business
Difficulty: Introductory | Prerequisites: Basic algebra and function notation (knowing what f(x) means, how to evaluate expressions, and how to work with fractions and exponents).
Limits and continuity are the foundation that the entire calculus course is built on. Before you can differentiate or integrate anything, you need to understand what it means for a function to approach a value and what it means for a function to have no breaks or jumps. This chapter introduces the language and the mechanics you will use every week from here on. If you are coming in cold, make sure you are comfortable with evaluating functions, working with piecewise definitions, and simplifying rational expressions.
A limit describes the value a function approaches as its input gets close to some number, even if the function is not defined there. Continuity means the function has no gaps: f(c) exists, the limit exists, and they are equal. These two ideas set up the definition of the derivative, which is introduced at the end of the chapter.
Limit
The value that f(x) approaches as x gets closer and closer to some number c, written lim (x→c) f(x). The function does not need to equal that value at c, or even be defined there.
In simple terms, it is the answer to: "What is f(x) heading towards as x gets near c?"
One-sided limit (left-hand limit)
The value f(x) approaches as x gets close to c from values less than c, written lim (x→c⁻) f(x).
Think of it as: "What happens to f(x) if I sneak up on c from the left side of the number line?"
One-sided limit (right-hand limit)
The value f(x) approaches as x gets close to c from values greater than c, written lim (x→c⁺) f(x).
Think of it as: "What happens if I approach c from the right?"
Two-sided limit
The overall limit lim (x→c) f(x). It exists if and only if the left-hand limit and the right-hand limit both exist and are equal to each other.
In simple terms, both sides have to agree, or the limit does not exist.
Continuity at a point
A function f is continuous at x = c when three conditions are all met: f(c) is defined, lim (x→c) f(x) exists, and lim (x→c) f(x) = f(c).
Think of it as: you can draw the graph through that point without lifting your pen.
DNE (Does Not Exist)
Used when a limit has no single finite value. This can happen because the left and right limits disagree, or because the function blows up to infinity.
In simple terms, the function is not settling on one answer.
Direct substitution
The simplest method: plug the value of c into f(x). If you get a real number, that is your limit.
Works whenever the function is a polynomial, or a rational function where the denominator is not zero at c.
Limits at infinity (x → ∞ or x → −∞)
Key rule: if A and k are constants with k > 0, then lim (x→∞) A / xᵏ = 0 and lim (x→−∞) A / xᵏ = 0.
In plain terms, any constant divided by a growing power of x shrinks to zero.
Examples: lim (x→∞) 1/x = 0, lim (x→∞) 1/x² = 0, lim (x→∞) 100/x = 0.
Rational functions at infinity
Divide every term in the numerator and denominator by the highest power of x that appears.
Example: lim (x→∞) (3x² + 3x + 1) / (3x² − 5x + 2). Divide everything by x². The lower-power terms go to zero, leaving 3/3 = 1.
Limits that blow up (infinite limits)
lim (x→0) 1/x² = +∞ (approaches positive infinity from both sides). The limit "does not exist" in the finite sense, but we say the function grows without bound.
lim (x→0) 1/x: from the left it goes to −∞, from the right it goes to +∞. The two sides disagree, so the limit DNE.
The core rule
The two-sided limit lim (x→c) f(x) exists if and only if both one-sided limits exist and are equal:
lim (x→c⁻) f(x) = lim (x→c⁺) f(x) = L, then lim (x→c) f(x) = L.
If the left and right values differ, or if either side does not settle on a finite value, the two-sided limit does not exist.
How to check (piecewise functions)
These questions almost always involve a piecewise-defined function, where the rule changes at some boundary value.
Compute the left-hand limit using the piece that applies for x < c.
Compute the right-hand limit using the piece that applies for x > c (or x ≥ c).
Compare: equal means the limit exists; not equal means DNE.
Worked example from the lecture
f(x) is defined piecewise. The lecture checks whether lim (x→1) f(x) exists by evaluating each side separately. Left-hand limit = A + 5 and right-hand limit = 1 − 3 + 4 = 2. Setting these equal leads to the value of the constant A (see the continuity section below).
Three conditions for continuity at x = c
f(c) is defined (the function produces a value at c).
lim (x→c) f(x) exists (the two-sided limit is a finite number).
lim (x→c) f(x) = f(c) (the limit equals the function value).
All three must hold. If any one fails, the function is not continuous at c.
Example: show f(x) = x + 1 is continuous at x = 3
f(3) = 3 + 1 = 4. The function is defined.
lim (x→3) (x + 1) = 4. The limit exists.
f(3) = 4 = lim (x→3) f(x). They match.
All three conditions satisfied, so f is continuous at x = 3.
Example: finding a constant for continuity
Given the piecewise function:
f(x) = Ax + 5 if x < 1
f(x) = x² − 3x + 4 if x ≥ 1
For f to be continuous at x = 1, the left and right limits must match and equal f(1).
Left-hand limit: lim (x→1⁻) (Ax + 5) = A + 5.
Right-hand limit (and f(1)): lim (x→1⁺) (x² − 3x + 4) = 1 − 3 + 4 = 2.
Set A + 5 = 2, so A = −3.
Limit at infinity for a constant over a power
\lim_{x \to \pm\infty} \frac{A}{x^k} = 0 \quad (k > 0)Two-sided limit existence condition
\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = LThree conditions for continuity at x = c
f(c) \text{ is defined}, \quad \lim_{x \to c} f(x) \text{ exists}, \quad \lim_{x \to c} f(x) = f(c)Limits underpin every marginal-cost and marginal-revenue calculation in business calculus. When an economist says "the cost of producing one more unit," they are describing a limit. Continuity matters whenever you model a process that should not have sudden jumps, such as a revenue function or a production schedule: a discontinuity signals a break in the model that needs explaining.
Students often think a limit equals the function value at that point. It does not have to. The limit is about what f(x) approaches, not what f(c) equals.
Students often assume that if lim (x→0) 1/x² = ∞, the limit "exists." In standard calculus, a limit of ±∞ means the limit does not exist as a finite number. Your professor may use the phrase "the limit is infinite" as shorthand, but on an exam "DNE" is usually the expected answer.
Students mix up one-sided and two-sided limits. Saying "the limit is 3 from the left" does not mean the overall limit is 3. You must check both sides.
Students forget to check all three conditions for continuity. Showing that the limit exists is not enough on its own; you also need f(c) to be defined and equal to that limit.
⚠️ "Find the value of the constant A so that f is continuous" is a classic exam question. Know the setup: set left limit = right limit = f(c), solve for A.
⚠️ You will be asked to evaluate limits at infinity by dividing by the highest power. This is quick marks if you know the technique.
⚠️ Expect at least one question asking you to determine whether a limit exists for a piecewise function, using one-sided limits.
⚠️ The three conditions for continuity are routinely tested as a "state the definition and verify" problem.
True or False: If lim (x→c⁻) f(x) = 5 and lim (x→c⁺) f(x) = 7, the two-sided limit exists. (False)
Fill in the blank: lim (x→∞) 100/x³ = ___. (0)
True or False: A function can have a limit at x = c even if f(c) is not defined. (True)
Fill in the blank: For f to be continuous at c, lim (x→c) f(x) must equal ___. (f(c))
True or False: lim (x→0) 1/x = ∞. (False, the left-hand limit is −∞ while the right-hand limit is +∞, so the limit DNE.)
Q: Evaluate lim (x→∞) (5x² + 2x) / (3x² − x + 1).
A: Divide every term by x². The expression becomes (5 + 2/x) / (3 − 1/x + 1/x²). As x → ∞, the fractions vanish, leaving 5/3.
Q: Determine whether lim (x→0) 1/x exists.
A: From the left, 1/x → −∞. From the right, 1/x → +∞. The one-sided limits are not equal, so the limit does not exist.
Q: Is f(x) = (x² − 9)/(x − 3) continuous at x = 3?
A: f(3) is undefined because the denominator is zero, so the first condition for continuity fails. The function is not continuous at x = 3. (Note: the limit does exist and equals 6, since (x² − 9)/(x − 3) simplifies to x + 3 for x ≠ 3.)
Q: Given f(x) = Ax + 5 for x < 1 and f(x) = x² − 3x + 4 for x ≥ 1, find A so that f is continuous everywhere.
A: Set the left-hand limit equal to the right-hand limit at x = 1. A(1) + 5 = (1)² − 3(1) + 4 gives A + 5 = 2, so A = −3.
Q: State the three conditions for continuity of f at x = c.
A: (1) f(c) is defined. (2) lim (x→c) f(x) exists. (3) lim (x→c) f(x) = f(c).
Limits feed directly into the definition of the derivative (Chapter 1, Part 2). The derivative is defined as a limit of a difference quotient, so everything you practise here with evaluating limits carries straight into differentiation. Continuity also matters later: the Intermediate Value Theorem (used in optimisation problems) requires the function to be continuous on the interval. In business calculus specifically, continuous functions model revenue, cost, and profit curves where sudden jumps would not make economic sense.
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