Difficulty: Introductory | Prerequisites: Basic algebra (factoring, simplifying rational expressions).
Limits are the gateway concept in calculus. They describe how a function behaves as the input approaches a particular value, even when the function is not defined there. This topic sits right at the start of a business calculus course because derivatives (coming next) are built entirely on the idea of a limit. You should already be comfortable factoring polynomials, simplifying rational expressions, and working with square roots before tackling this material.
A limit tells you what value a function is heading towards as x gets close to some number, even if the function never actually lands there. You evaluate limits by direct substitution when possible, and by algebraic tricks (factoring, rationalising) when substitution gives you 0/0. For polynomials at infinity, only the highest-power term matters.
Limit
The value that f(x) approaches as x approaches a particular number c. Written as lim (x→c) f(x) = L.
In simple terms, it is the number the output is getting closer and closer to, even if the function never reaches it.
Direct substitution
Evaluating a limit by plugging the target x-value straight into the function.
Think of it as the first thing you try: just put the number in and see what happens.
Indeterminate form (0/0)
The result you get when direct substitution produces 0 in both the numerator and denominator. This does not mean the limit is zero or undefined; it means you need to do more algebra.
In simple terms, 0/0 is a signal to keep working, not an answer.
Factoring and cancelling
Rewriting a rational expression so that the factor causing the 0/0 appears in both numerator and denominator, then cancelling it before substituting.
Rationalising (conjugate multiplication)
Multiplying numerator and denominator by the conjugate of a square-root expression to eliminate the radical and resolve an indeterminate form.
Limit at infinity
The behaviour of f(x) as x grows without bound (x→∞) or decreases without bound (x→ −∞). For rational functions, compare the degrees of numerator and denominator. For polynomials, the leading term decides.
Think of it as asking: "What does this function do in the long run?"
Correct limit notation
The proper way to write a limit calculation. You must keep lim (x→c) in front of every expression until you substitute; dropping it early is the most common notation error.
DNE (Does Not Exist)
Used when a two-sided limit does not settle on a single finite value, typically because the left-hand and right-hand limits disagree or the function blows up to infinity.
Writing limits correctly is worth marks on its own. The rule is simple: the "lim" symbol must stay attached to every algebraic expression until you perform the final substitution. Dropping it early is marked "IC" (incorrect) even if your arithmetic is right.
What correct notation looks like
Write lim (x→c) in front of the original expression.
Carry lim (x→c) through every simplification step.
Only drop the lim symbol at the step where you substitute the value of x.
Example of correct (C) notation
lim (x→3) (x + 4) = 3 + 4 = 7. The limit symbol is present until substitution happens. This is correct.
Example of incorrect (IC) notation, then fixed
IC version: lim (x→−1) 5/(x+3) = lim (x→−1) 5/(−1+3) = 5/2. The error is keeping the lim symbol after substitution has already been performed.
Fixed version: lim (x→−1) 5/(x+3) = 5/(−1+3) = 5/2. The lim symbol drops at the same step as the substitution.
Another common IC pattern
IC version: lim (x→−2) (x−2)/((x+2)(x−2)) = 1/(x+2) = 1/4. The lim symbol was dropped before the substitution step.
Fixed version: lim (x→−2) (x−2)/((x+2)(x−2)) = lim (x→−2) 1/(x+2) = 1/(−2+2). Here the lim symbol is carried through the cancellation step and only dropped at substitution.
Limits at infinity, notation
lim (x→∞) (2x² − 3x)/(3x² − 7). The correct approach: simplify to lim (x→∞) 2x²/3x² = lim (x→∞) 2/3 = 2/3. The lim symbol stays present through every simplification.
Strategy 1: Direct substitution
Always try this first. Plug the target value of x into f(x). If you get a real number, that is the limit.
Example: lim (x→−4) (−10) = −10. A constant function always returns the same value, so the limit is just that constant.
Example: lim (x→−3) (x+4)/(2x−5) = (−3+4)/(2(−3)−5) = 1/(−11) = −1/11.
Strategy 2: Factor, cancel, then substitute (resolving 0/0)
When direct substitution gives 0/0, factor both numerator and denominator to find the common factor causing the problem. Cancel it, then substitute.
Example: lim (x→2) (x²+2x−8)/(x²−4).
Substituting gives (4+4−8)/(4−4) = 0/0.
Factor: (x+4)(x−2) / (x+2)(x−2).
Cancel (x−2): lim (x→2) (x+4)/(x+2) = 6/4 = 3/2.
The cancelled factor is what made the function undefined at x = 2, but the limit still exists because the function's behaviour near x = 2 is well-defined.
Strategy 3: Rationalise using the conjugate
When a square root appears and substitution gives 0/0, multiply numerator and denominator by the conjugate of the radical expression.
Example: lim (x→36) (x−36)/(√x−6).
Substituting gives 0/0.
Multiply top and bottom by (√x+6): (x−36)(√x+6) / ((√x−6)(√x+6)) = (x−36)(√x+6) / (x−36).
Cancel (x−36): lim (x→36) (√x+6) = √36+6 = 12.
Strategy 4: Limits at infinity for polynomials
For a polynomial, only the leading term (highest-degree term) matters as x→∞ or x→−∞.
Example: lim (x→−∞) (x³−1)(2x²+5). The dominant behaviour comes from x³ · 2x² = 2x⁵. As x→−∞, 2x⁵ → −∞.
Example: lim (x→∞) (3x⁴−x⁵+2). The leading term is −x⁵. As x→∞, −x⁵ → −∞.
Strategy 5: Limits at infinity for rational functions
Compare the degree of the numerator and denominator:
Same degree: the limit equals the ratio of the leading coefficients.
Numerator degree higher: the limit is ±∞.
Denominator degree higher: the limit is 0.
Example: lim (x→∞) (2x²−3x)/(3x²−7) = 2/3. Both numerator and denominator have degree 2, so the answer is 2/3.
Difference of squares (used constantly in factoring for limits)
a² − b² = (a + b)(a − b)
Conjugate multiplication (for rationalising)
Multiply by (√x + a)/(√x + a) to convert (√x − a) in the denominator into x − a².
Rational function limit at infinity
For f(x) = (aₙxⁿ + ...)/(bₘxᵐ + ...):
If n = m: limit = aₙ/bₘ
If n > m: limit = ±∞
If n < m: limit = 0
Polynomial limit at infinity
Only the leading term matters. Determine its sign by considering both the coefficient and whether x→+∞ or x→−∞ (odd powers flip sign with negative x, even powers do not).
Limits are how economists and analysts describe marginal behaviour: the cost of producing one more unit, the revenue from one more customer, the rate of change at a specific moment. Every derivative in business calculus is defined as a limit, so the algebra you practise here is the exact algebra you will use when computing derivatives of cost, revenue, and profit functions.
Students often think 0/0 means the limit is zero or undefined. It does not. 0/0 is an indeterminate form, which means more work is needed. The limit could be any finite number.
Students sometimes drop the lim symbol too early and then keep simplifying without it. Your professor will mark this as incorrect notation even if the final number is right.
When rationalising, students frequently forget to multiply both the numerator and the denominator by the conjugate. You must multiply both to keep the expression equivalent.
For limits at infinity, students sometimes try to "plug in infinity." Infinity is not a number. Instead, identify the dominant term and reason about its behaviour.
⚠️ Notation questions ("Is this correct or incorrect? Fix it.") appear on exams. They test whether you understand that the lim symbol is not optional decoration.
⚠️ 0/0 problems are near-guaranteed on every exam. Know your three resolution strategies: factor and cancel, rationalise, or simplify.
⚠️ Limits at infinity for rational functions: the degree-comparison rule is quick marks if you memorise it.
⚠️ Polynomial limits at infinity: identify the leading term, account for the sign, and state whether the limit is +∞ or −∞.
True or False: If direct substitution gives 0/0, the limit does not exist.
False. 0/0 is indeterminate; further algebra is needed to find the actual limit.
Fill in the blank: When evaluating lim (x→∞) for a rational function where numerator and denominator have the same degree, the limit equals ________.
The ratio of the leading coefficients.
True or False: You may drop the lim symbol at any step during simplification.
False. Keep it until the step where you substitute the value of x.
Fill in the blank: To rationalise √x − 6 in a denominator, multiply top and bottom by ________.
√x + 6 (the conjugate).
Q: Evaluate lim (x→−3) (x+4)/(2x−5). Show correct notation.
A: lim (x→−3) (x+4)/(2x−5) = (−3+4)/(2(−3)−5) = 1/(−11) = −1/11.
Q: Evaluate lim (x→2) (x²+2x−8)/(x²−4).
A: Direct substitution gives 0/0. Factor: (x+4)(x−2)/((x+2)(x−2)). Cancel (x−2). Then lim (x→2) (x+4)/(x+2) = 6/4 = 3/2.
Q: Evaluate lim (x→36) (x−36)/(√x−6).
A: Direct substitution gives 0/0. Multiply by the conjugate (√x+6)/(√x+6). This gives (x−36)(√x+6)/(x−36). Cancel (x−36). Then lim (x→36) (√x+6) = 6+6 = 12.
Q: Evaluate lim (x→−∞) (x³−1)(2x²+5).
A: The dominant term is 2x⁵. As x→−∞, x⁵ is negative (odd power), so 2x⁵→−∞. The limit is −∞.
Q: Evaluate lim (x→∞) (3x⁴−x⁵+2).
A: The leading term is −x⁵. As x→∞, −x⁵→−∞. The limit is −∞.
This connects directly to derivatives, which are defined as lim (h→0) [f(x+h) − f(x)]/h. Every derivative you compute later in this course is a limit in disguise, and the 0/0 resolution techniques you learn here (factoring, cancelling) are the same ones used to derive the power rule and other differentiation formulas.
Limits at infinity connect to the behaviour of cost, revenue, and profit functions as production scales up, a recurring theme in business calculus applications.
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