Source: Lecture notes #21, Texas A&M University
Tags: limits at infinity, vertical asymptote, horizontal asymptote, continuity, continuous function, discontinuity, intermediate value theorem, IVT, calculus 1, CALC 157
Limits at infinity describe what happens to a function as x grows very large or very small, and they connect directly to vertical and horizontal asymptotes. Continuity means a function has no breaks, holes, or jumps at a point. The Intermediate Value Theorem guarantees that a continuous function on a closed interval hits every y-value between its endpoints.
Vertical asymptote
A vertical line x = c where the function heads towards positive or negative infinity. Occurs when a non-zero number is divided by zero (not 0/0).
Horizontal asymptote
A horizontal line y = L that the function approaches as x → ∞ or x → –∞. Found by comparing the degrees of the numerator and denominator.
Continuous function
A function with no breaks, holes, or jumps. At every point c in its domain, three conditions hold: the limit exists, f(c) is defined, and the limit equals f(c).
Discontinuity (types)
Infinite discontinuity: a break in the graph (vertical asymptote).
Point discontinuity: a hole in the graph.
Jump discontinuity: the function jumps from one value to another.
Intermediate Value Theorem (IVT)
If f(x) is continuous on a closed interval [a, b], then for every value d between f(a) and f(b), there exists at least one c between a and b such that f(c) = d.
Vertical asymptotes:
A vertical asymptote appears at x = c when the denominator equals zero but the numerator does not.
The key indicator is a non-zero number divided by zero.
Example: f(x) = (2x + 1) / (x – 3) has a vertical asymptote at x = 3.
lim (2x + 1)/(x – 3) as x → 3⁺ = +∞
lim (2x + 1)/(x – 3) as x → 3⁻ = –∞
Horizontal asymptotes (end behaviour):
Found by taking the limit as x → ∞ (or x → –∞).
Compare the highest-degree terms in the numerator and denominator.
Example: lim (2x + 1)/(x – 3) as x → ∞
Highest-degree terms: 2x / x = 2
Horizontal asymptote: y = 2
Degree comparison rules for rational functions:
Degree of numerator < degree of denominator: horizontal asymptote at y = 0.
Degree of numerator = degree of denominator: horizontal asymptote at the ratio of leading coefficients.
Degree of numerator > degree of denominator: no horizontal asymptote (limit DNE, function grows without bound).
Important note: When a limit is infinite (±∞), we technically say the limit does not exist. Infinity is not a number, so the limit DNE. Saying "the limit is ∞" is a way of describing the behaviour, not assigning a value.
A function is continuous at a point x = a if all three conditions are met:
lim f(x) as x → a exists (left and right limits agree).
f(a) is defined (no hole or undefined expression).
lim f(x) as x → a = f(a) (the limit matches the actual function value).
Types of discontinuity:
Infinite discontinuity: vertical asymptote, the function blows up.
Point discontinuity (hole): the limit exists but f(a) is either undefined or different from the limit.
Jump discontinuity: the left-hand and right-hand limits exist but disagree, so the general limit DNE.
A continuous function has:
No breaks in the graph (no infinite discontinuity).
No holes (no point discontinuity).
No jumps (no jump discontinuity).
No undefined points or vertical asymptotes in the interval you care about.
Applies only to functions that are continuous on a closed interval [a, b].
States: for every value d between f(a) and f(b), there is at least one c in (a, b) where f(c) = d.
In plain terms, a continuous function cannot skip over a y-value. If the function is at 3 when x = 1 and at 7 when x = 5, it must pass through every value between 3 and 7 somewhere in that interval.
Practical use:
Often used to prove that a root (zero) exists. If f(a) is negative and f(b) is positive (or vice versa), and f is continuous on [a, b], there must be at least one c where f(c) = 0.
Referenced in the notes as a "big use" for "the fat project," suggesting it appears in applied coursework.
Vertical asymptote test:
If f(x) = p(x) / q(x) and q(c) = 0 while p(c) ≠ 0, then x = c is a vertical asymptote.
Horizontal asymptote (equal degrees):
lim [aₙxⁿ + ...] / [bₙxⁿ + ...] as x → ∞ = aₙ / bₙ
Continuity conditions at x = a:
lim f(x) as x → a exists
f(a) is defined
lim f(x) as x → a = f(a)
Intermediate Value Theorem:
If f is continuous on [a, b] and d is between f(a) and f(b), then there exists c ∈ (a, b) such that f(c) = d.
⚠️ A non-zero number divided by zero signals a vertical asymptote. 0/0 signals a hole (use factoring or conjugate to resolve).
⚠️ The degree comparison shortcut for horizontal asymptotes is one of the fastest marks on an exam. Know all three cases cold.
⚠️ "Limit equals infinity" technically means the limit does not exist. Infinity is a description of behaviour, not a value.
⚠️ Continuity questions almost always test whether you know all three conditions, not just one. A common mistake is forgetting to check that f(a) is defined.
⚠️ The IVT requires continuity on a closed interval. If the function is not continuous, the theorem does not apply, even if it looks like it should.
⚠️ The IVT tells you a value exists but not what it is or where exactly it occurs.
Q: How do you identify a vertical asymptote from a rational function?
A: Set the denominator equal to zero and check that the numerator is not also zero at that point. If the numerator is non-zero, there is a vertical asymptote at that x-value.
Q: What is the horizontal asymptote of f(x) = (2x + 1) / (x – 3)?
A: The numerator and denominator have the same degree (both degree 1). The horizontal asymptote is the ratio of leading coefficients: 2/1 = 2, so y = 2.
Q: If the numerator's degree is less than the denominator's, what is the horizontal asymptote?
A: y = 0. The denominator grows faster, so the function shrinks towards zero.
Q: State the three conditions for continuity at x = a.
A: The limit as x → a must exist, f(a) must be defined, and the limit must equal f(a).
Q: What is the difference between a hole and a vertical asymptote?
A: A hole occurs at a point where both the numerator and denominator are zero (0/0 form, cancellable). A vertical asymptote occurs where the denominator is zero but the numerator is not (function blows up to ±∞).
Q: A continuous function on [1, 5] has f(1) = –3 and f(5) = 4. Must there be a root between 1 and 5?
A: Yes. By the Intermediate Value Theorem, since f is continuous and f(1) is negative while f(5) is positive, there must be at least one c in (1, 5) where f(c) = 0.
Q: Can the IVT be used if the function has a discontinuity in the interval?
A: No. The IVT requires the function to be continuous on the entire closed interval [a, b]. A discontinuity anywhere in that interval invalidates the theorem.
limits at infinity, infinite limits, vertical asymptote, horizontal asymptote, end behaviour, degree comparison, leading coefficient ratio, continuous function, continuity, three conditions for continuity, discontinuity types, infinite discontinuity, point discontinuity, jump discontinuity, removable discontinuity, Intermediate Value Theorem, IVT, existence of roots, zero of a function, calculus 1, CALC 157, Texas A&M, MATH 151