Difficulty: Introductory
Prerequisites: Algebra II (factoring polynomials, rational expressions, function notation). If you are shaky on factoring trinomials or simplifying fractions with variables, review those first.
Big Picture
Module 1 is the gateway to everything else in MAC 2233. Limits formalise the idea of "what value does a function approach?" and that idea underpins every derivative and integral you will meet later. Rates of change connect limits to real business questions: how fast is cost growing, how quickly is revenue shifting. If you understand limits and the difference quotient here, Modules 2 and 3 will feel like natural extensions rather than new topics.
TL;DR
A limit describes the value a function approaches as x gets close to some number, even if the function is undefined there. You evaluate limits by direct substitution, factoring, or inspecting one-sided behaviour. The average rate of change formula (the difference quotient) measures how fast a function's output changes between two x-values, and it leads directly into derivatives.
Limit
The value a function f(x) approaches as x gets arbitrarily close to a given number, without necessarily reaching it. In simple terms, it is the answer to "where is this function heading?"
Continuous function
A function that has no breaks, holes, or jumps at a given point. If you can draw it through that point without lifting your pen, it is continuous there. Formally, f is continuous at x = a when lim(x→a) f(x) = f(a).
Restriction (of a function)
An x-value that makes the function undefined, typically because it produces division by zero. These are the values you must watch for before plugging in.
One-sided limit
The value a function approaches from only one direction. lim(x→a⁺) means approaching from the right (values slightly larger than a); lim(x→a⁻) means approaching from the left (values slightly smaller than a).
Average rate of change
The ratio [f(x + h) - f(x)] / h, measuring how much the output changes per unit change in x over an interval. Think of it as the slope of the line connecting two points on the curve.
Difference quotient
Another name for the average rate of change formula. This expression is the foundation of the derivative, which you will meet in Module 2.
The central idea: a limit asks what value f(x) is heading toward as x approaches some target, not what f(x) equals at that target.
Step 1: Check for restrictions. Look at the denominator. Any x-value that makes the denominator zero is a restriction. For example, f(x) = (x + 4) / (x² - 4) is undefined at x = -2 and x = 2, because x² - 4 = (x + 2)(x - 2) = 0 at those points.
Step 2: Try direct substitution. Plug the target x-value into the function. If you get a real number (no division by zero), that number is the limit.
Step 3: If substitution gives 0/0, factor and simplify. Cancel the common factor, then substitute again.
A function like f(x) = (5x + 2) / (x² + 16) has no real restrictions, because x² + 16 is always positive. It is continuous on all real numbers.
When direct substitution produces 0/0, factoring is your main tool.
Simple rational functions (no 0/0). Example: lim(x→1) of (2x + 3)/(x + 1). Substitution gives (2(1) + 3)/(1 + 1) = 5/2. Done.
Factorable 0/0 cases. Example: lim(x→2) of (2x² + 9x + 10)/(x² - x - 6).
Numerator: 2x² + 9x + 10. Multiply A and C: 2 × 10 = 20. Find two numbers that multiply to 20 and add to 9: that is 5 and 4. Rewrite the middle term: 2x² + 4x + 5x + 10 = 2x(x + 2) + 5(x + 2) = (2x + 5)(x + 2).
Denominator: x² - x - 6. Multiply A and C: 1 × (-6) = -6. Find two numbers that multiply to -6 and add to -1: that is -3 and 2. Rewrite: x² - 3x + 2x - 6 = x(x - 3) + 2(x - 3) = (x + 2)(x - 3).
Cancel the common factor (x + 2): result is (2x + 5)/(x - 3).
Substitute x = 2: (2(2) + 5)/(2 - 3) = 9/(-1) = -9.
One-sided limits
lim(x→a⁺) means x approaches a from the right (e.g. x = 3.1, 3.01, 3.001 approaching 3).
lim(x→a⁻) means x approaches a from the left (e.g. x = 2.9, 2.99, 2.999 approaching 3).
To evaluate, substitute a value slightly to the right or left and check the sign of the result. If the output grows without bound, the one-sided limit is +∞ or -∞.
Example: lim(x→-3⁻) of (3x + 2)/(x + 3). Try x = -3.1: numerator = 3(-3.1) + 2 = -7.2 (negative). Denominator = -3.1 + 3 = -0.1 (negative). Negative/negative = positive, and the denominator is shrinking toward zero, so the limit is +∞.
Limits at infinity
When finding lim(x→∞) or lim(x→-∞) of a rational function, compare the highest-power terms in the numerator and denominator.
Keep only the leading terms and simplify. Everything else becomes negligible.
Example: lim(x→-∞) of (9000x - 5x² - 5x⁴) / (4x² + 4x). The leading terms are -5x⁴ (numerator) and 4x² (denominator). The ratio simplifies to -5x²/4. As x→-∞, x² is positive (even power), so the expression heads toward -∞.
Sign rule for powers of negative x: even exponent produces a positive result, odd exponent produces a negative result.
\text{Average rate of change} = \frac{f(x+h) - f(x)}{h}where x is the starting point and h is the distance to the second point (h = x₂ - x₁).
\lim_{x \to a} f(x) = Lmeans f(x) approaches L as x approaches a.
For limits at infinity of a rational function P(x)/Q(x), divide numerator and denominator by the highest power of x in the denominator, then let x→∞.
The average rate of change measures how fast f(x) changes over an interval. The formula is [f(x + h) - f(x)] / h.
x is the reference point (the starting x-value).
h is the gap between the two x-values (h = x₂ - x₁).
Worked example: general rate of change
Given f(x) = -2x - 2x², find the average rate of change formula.
Compute f(x + h): -2(x + h) - 2(x + h)² = -2x - 2h - 2(x² + 2xh + h²) = -2x - 2h - 2x² - 4xh - 2h².
Subtract f(x): (-2x - 2h - 2x² - 4xh - 2h²) - (-2x - 2x²) = -2h - 4xh - 2h².
Divide by h: h(-2 - 4x - 2h) / h = -2 - 4x - 2h.
The average rate of change simplifies to -2 - 4x - 2h.
Worked example: specific interval
Using the same function, find the rate of change from x = -2 to x = -1.8.
Here x = -2 and h = -1.8 - (-2) = 0.2.
Substitute into -2 - 4x - 2h: -2 - 4(-2) - 2(0.2) = -2 + 8 - 0.4 = 5.6.
The average rate of change over that interval is 5.6.
The average rate of change is how businesses measure growth or decline over a period. If C(x) is the cost of producing x units, then [C(x + h) - C(x)] / h tells you the average cost per additional unit over that range. This is exactly how a company estimates whether scaling production is worth the expense.
Students often think the limit of f(x) at x = a must equal f(a). It does not. The limit is about where the function is heading, not where it actually is. A function can have a hole at x = 2 and still have a perfectly defined limit there.
Students confuse restrictions with limits. A restriction tells you the function is undefined; a limit tells you what value the function approaches near that point. They are separate questions.
When factoring to resolve 0/0, students sometimes forget to cancel before substituting. If you substitute into the unfactored form, you will get 0/0 again and think the limit does not exist.
For limits at infinity, students sometimes keep all terms instead of isolating the leading terms. Only the highest-power terms matter; everything else vanishes as x grows.
⚠️ Factoring trinomials to resolve 0/0 indeterminate forms is heavily tested. Practise the AC method (multiply A and C, find the pair, split the middle term, factor by grouping) until it is automatic.
⚠️ One-sided limits and sign analysis (positive/negative of numerator and denominator) appear in questions about continuity and vertical asymptotes.
⚠️ The difference quotient (average rate of change) is the bridge between Module 1 and Module 2. Expect at least one problem that asks you to derive the simplified rate-of-change expression and then evaluate it at specific values.
⚠️ Limits at infinity: know how to identify the leading terms and determine the sign of the result. The even/odd exponent sign rule for negative x is a common trap.
True or False: If f(2) is undefined, then lim(x→2) f(x) does not exist.
False. The limit can exist even when the function is undefined at that point.
Fill in the blank: To evaluate lim(x→a) of a rational function, the first thing to try is ______.
Direct substitution.
True or False: When direct substitution gives 0/0, the limit does not exist.
False. 0/0 is an indeterminate form, meaning you need to simplify further (usually by factoring).
Fill in the blank: In the rate of change formula [f(x+h) - f(x)] / h, the variable h represents ______.
The difference between the two x-values.
True or False: For lim(x→-∞), an even power of x produces a negative result.
False. An even power of x is always positive, regardless of the sign of x.
Q: Evaluate lim(x→1) of (2x + 3)/(x + 1).
A: Direct substitution gives (2(1) + 3)/(1 + 1) = 5/2.
Q: Evaluate lim(x→2) of (2x² + 9x + 10)/(x² - x - 6).
A: Factor numerator to (2x + 5)(x + 2) and denominator to (x + 2)(x - 3). Cancel (x + 2). Substitute x = 2 into (2x + 5)/(x - 3) = 9/(-1) = -9.
Q: Given f(x) = -2x - 2x², find the average rate of change from x = -2 to x = -1.8.
A: h = 0.2. The simplified rate of change is -2 - 4x - 2h. Substituting: -2 - 4(-2) - 2(0.2) = -2 + 8 - 0.4 = 5.6.
Q: Evaluate lim(x→-3⁻) of (3x + 2)/(x + 3). State whether the limit is +∞ or -∞.
A: As x→-3 from the left, the numerator approaches -7 (negative) and the denominator approaches 0 from the negative side. Negative divided by negative is positive and grows without bound, so the limit is +∞.
Q: Find lim(x→-∞) of (9000x - 5x² - 5x⁴)/(4x² + 4x).
A: Leading terms: -5x⁴ / 4x² = -5x²/4. As x→-∞, x² is positive, so the expression approaches -∞.
The difference quotient from this module becomes the derivative in Module 2. Where Module 1 computes the average rate of change over an interval, Module 2 shrinks that interval to zero and finds the instantaneous rate. Every differentiation rule (power, product, quotient, chain) is a shortcut for the limit process you learnt here.
Continuity and one-sided limits reappear in Module 3 when you determine whether a function has a maximum or minimum at a point, and when you analyse the behaviour of cost, revenue, and demand functions.
limits, evaluating limits, limit definition, factoring to find limits, indeterminate form, 0/0, direct substitution, one-sided limits, left-hand limit, right-hand limit, limit at infinity, leading term analysis, average rate of change, difference quotient, rate of change formula, continuity, continuous function, restrictions, vertical asymptote, MAC 2233, calculus for business, FIU, Professor Fox