Difficulty: Intermediate | Prerequisites: Basic limits (Section 2.1), polynomial and rational functions.
This section builds on the basic idea of a limit by asking: what happens to a function's output when the input grows without bound, or when the input approaches a value that makes the function blow up? These are the two sides of asymptotic behaviour. Horizontal asymptotes describe the long-run output of a function (what y approaches as x heads off to infinity), while vertical asymptotes describe the input values where the function's output explodes toward infinity. You need a solid grip on polynomial degree and how to factor denominators before this will click.
A horizontal asymptote is the y-value a function settles toward as x goes to positive or negative infinity. A vertical asymptote is an x-value where the function's output shoots off to infinity. For rational functions, you can find both by comparing the degrees of the numerator and denominator and by factoring the denominator.
Horizontal asymptote
A horizontal line y = c that the graph of a function approaches as x tends to positive or negative infinity. Formally, if lim (x→∞) f(x) = c, then y = c is a horizontal asymptote.
In simple terms, it is the value y is trying to reach when x gets extremely large (or extremely negative).
Vertical asymptote
A vertical line x = a where the function's output increases or decreases without bound. If lim (x→a⁺) f(x) = ∞ or −∞, then x = a is a vertical asymptote.
Think of it as a wall the graph races toward but never crosses (in the neighbourhood of that x-value).
Slant (oblique) asymptote
A diagonal line that the graph approaches when the degree of the numerator is exactly one more than the degree of the denominator.
In simple terms, it is what happens when the function does not level off to a flat line but instead tilts toward a sloped line as x grows.
Rational function
A function of the form f(x) = n(x) / d(x), where n(x) and d(x) are polynomials.
Think of it as a fraction where both the top and bottom are polynomials.
Limit at infinity
The value that f(x) approaches as x grows without bound (x→∞) or decreases without bound (x→−∞).
In simple terms, it answers the question: where does the graph end up if you scroll far enough to the right (or left)?
Infinite limit
A limit where the function's output itself grows without bound (heads to ∞ or −∞) as x approaches some finite value.
In simple terms, the function blows up near a particular x-value rather than settling to a number.
Horizontal asymptote (HA): appears when the function approaches a constant y-value as x→±∞. Identified by evaluating the limit at infinity.
Vertical asymptote (VA): appears at x = a when the denominator of a rational function equals zero at x = a and the numerator does not equal zero at x = a. Found by factoring the denominator and solving d(x) = 0, then checking that n(a) ≠ 0.
Slant asymptote: appears when the numerator's degree is exactly one higher than the denominator's degree. Found by polynomial long division. (Not the focus of this warmup, but named in Problem 1.)
Given f(x) = (aₘxᵐ + ... + a₁x + a₀) / (bₙxⁿ + ... + b₁x + b₀), where m is the degree of the numerator and n is the degree of the denominator:
Case 1: m < n (numerator degree smaller). The horizontal asymptote is y = 0. The denominator grows faster than the numerator, so the fraction shrinks to zero.
Case 2: m = n (degrees equal). The horizontal asymptote is y = aₘ / bₙ, the ratio of the leading coefficients. Both top and bottom grow at the same rate, so everything cancels except that ratio.
Case 3: m > n (numerator degree larger). There is no horizontal asymptote. The numerator dominates and the function's output grows without bound.
Set the denominator equal to zero and solve.
Factor first if possible. For example, x² − 7x + 6 = (x − 1)(x − 6), giving vertical asymptotes at x = 1 and x = 6.
Confirm that the numerator is not also zero at the same x-value. If both numerator and denominator are zero, you may have a hole rather than an asymptote.
f(x) = (3x² + 2x + 1) / (2x² − 4): degrees equal (both 2), so HA is y = 3/2.
g(x) = (4x⁷ − 5x³ − 3) / (x⁵ − 1): numerator degree 7 > denominator degree 5, so no HA. The dominant term behaves like 4x², confirming unbounded growth.
h(x) = (x³ + 4x² − 5x + 2) / (x⁴ + 3x² − 1): numerator degree 3 < denominator degree 4, so HA is y = 0.
f(x) = (6x − 3x²) / (x² − 7x + 6): degrees equal (both 2), leading coefficients −3 and 1, so HA is y = −3. Denominator factors as (x − 1)(x − 6), giving VAs at x = 1 and x = 6.
C(t) = −1.5t² + 7.2t + 13.7: a polynomial (no denominator to create asymptotes), so no VA and no HA.
g(x) = (x³ + 5x) / (2x² − 8): numerator degree 3 > denominator degree 2, so no HA. Denominator factors as 2(x + 2)(x − 2), giving VAs at x = −2 and x = 2.
\text{If } m < n: \quad y = 0\text{If } m = n: \quad y = \frac{a_m}{b_n}\text{If } m > n: \quad \text{no horizontal asymptote}To find vertical asymptotes of f(x) = n(x) / d(x):
Solve d(x) = 0.
For each solution x = a, check that n(a) ≠ 0.
Each x = a that passes both checks is a vertical asymptote.
Horizontal asymptotes show up in business whenever a quantity levels off: average cost per unit approaches the variable cost as production volume grows, because the fixed-cost portion gets spread thinner and thinner. Vertical asymptotes appear in models where a denominator hits zero, such as pricing functions that blow up as supply approaches zero.
Students often think a graph can never cross a horizontal asymptote. It can. A horizontal asymptote describes end behaviour (as x→±∞), not what happens in the middle of the graph.
Students sometimes set the entire function equal to zero to find vertical asymptotes. Only the denominator is set to zero, and you must confirm the numerator is not also zero at that point.
Students confuse "no horizontal asymptote" with "no asymptote at all." When the numerator's degree exceeds the denominator's by exactly one, there is a slant asymptote instead.
Students occasionally forget that polynomials (no denominator) cannot have vertical or horizontal asymptotes. A function like C(t) = −1.5t² + 7.2t + 13.7 has neither.
⚠️ The three-case rule for horizontal asymptotes (m < n, m = n, m > n) is almost certainly on the exam. Be able to state each case and apply it quickly.
⚠️ Expect at least one problem that gives you a rational function and asks for both vertical and horizontal asymptotes. Factor the denominator and compare degrees: do both steps, in that order.
⚠️ Watch for a non-rational function (a plain polynomial or a root function) thrown in to test whether you recognise that it has no asymptotes at all.
True or false: If the degree of the numerator equals the degree of the denominator, the horizontal asymptote is y = 0. (False: y = leading coefficient ratio.)
Fill in the blank: A vertical asymptote occurs at x = a when d(a) = ____ and n(a) ≠ ____. (0; 0)
True or false: A polynomial function can have a vertical asymptote. (False.)
True or false: A graph can cross its horizontal asymptote. (True.)
Fill in the blank: When m > n for a rational function, there is no ____ asymptote. (horizontal)
Q: Find the horizontal asymptote of f(x) = (5x³ + 2x) / (10x³ − x² + 4).
A: Degrees are equal (both 3). HA is y = 5/10 = 1/2.
Q: Find the vertical asymptotes of g(x) = (x + 3) / (x² − 9).
A: Factor the denominator: x² − 9 = (x − 3)(x + 3). Setting each factor to zero gives x = 3 and x = −3. At x = −3 the numerator is also zero, so that is a hole, not a VA. The only vertical asymptote is x = 3.
Q: Does h(x) = 4x⁴ + x − 7 have any asymptotes?
A: No. h(x) is a polynomial with no denominator, so it has no vertical or horizontal asymptotes.
Q: Find all asymptotes of p(x) = (2x² − 8) / (x² − 5x + 6).
A: Degrees equal (both 2), so HA is y = 2/1 = 2. Factor the denominator: (x − 2)(x − 3). Check the numerator at x = 2: 2(4) − 8 = 0, so x = 2 is a hole, not a VA. Check the numerator at x = 3: 2(9) − 8 = 10 ≠ 0, so x = 3 is a vertical asymptote.
Q: A rational function has numerator degree 4 and denominator degree 2. What can you say about horizontal asymptotes?
A: Since m > n, there is no horizontal asymptote. The function grows without bound as x→±∞.
This connects to Section 2.1 (basic limits) because horizontal asymptotes are just limits evaluated at infinity rather than at a finite point. It also connects to curve sketching later in the course: knowing the asymptotes gives you the skeleton of the graph before you plot any points. In business calculus specifically, average cost functions are rational functions whose horizontal asymptotes represent the long-run cost per unit.
Horizontal asymptote, vertical asymptote, slant asymptote, oblique asymptote, limit at infinity, infinite limit, end behaviour, rational function, degree of polynomial, leading coefficient, asymptotic behaviour, curve sketching, holes in rational functions, removable discontinuity, Section 2.2, Calculus for Business, FIU, warmup exercises, limits and asymptotes