Course: Advanced Calculus I (University of Central Florida)
Source: Lecture notes on limits and continuity Difficulty: Introductory Prerequisites: Basic algebra, familiarity with function notation and graphing.
Tags: limits, continuity, limit notation, left-hand limit, right-hand limit, continuous function, discontinuity, limit laws, epsilon-delta, calculus foundations, lim x approaches c
Limits and continuity are the bedrock of calculus. Nearly every major concept that follows, from derivatives to integrals, is built on top of the idea of a limit. If you are joining late or coming back to this material after a break, this is where to start: without a solid grasp of limits, the rest of the course will feel like guesswork. Continuity, in turn, is the formal way of saying a function "behaves nicely" at a point, and it depends directly on the limit concept.
A limit describes the value a function approaches as its input gets close to some target value. Continuity at a point means the function is defined there, the limit exists, and the limit equals the function's value. Together, these two ideas underpin derivatives, integrals, and most of what follows in a calculus course.
Limit
The value that f(x) approaches as x gets closer and closer to a particular value c. Written as lim (x→c) f(x) = L. Think of it as: "What does the output home in on as the input sneaks up on c?"
Left-hand limit
The value f(x) approaches as x approaches c from values less than c (from the left on a number line). In simple terms, this means you are only looking at what happens to f(x) when x creeps up toward c from below.
Right-hand limit
The value f(x) approaches as x approaches c from values greater than c (from the right). Think of it as the mirror image of the left-hand limit, approaching from the other direction.
Existence of a limit
A limit exists at x = c only when the left-hand limit and the right-hand limit both exist and are equal. If they disagree, the limit does not exist at that point.
Continuity (at a point)
A function is continuous at a point c when three conditions hold: (1) f(c) is defined, (2) lim (x→c) f(x) exists, and (3) lim (x→c) f(x) = f(c). In simple terms, you can draw through the point without lifting your pen, and the function value matches what the limit predicts.
Continuous function
A function that is continuous at every point in its domain. Think of it as a function with no holes, jumps, or vertical asymptotes anywhere it is supposed to exist.
Discontinuity
A point where one or more of the three continuity conditions fails. The function might be undefined, the limit might not exist, or the limit and the function value might not match.
The notation lim (x→c) f(x) = L means that as x gets arbitrarily close to c, the output f(x) gets arbitrarily close to L.
The value of f at c itself does not matter for the limit. What matters is the behaviour of f(x) near c.
For the limit to exist, the function must approach the same value L from both the left and the right.
Consider f(x) = (x² − 1)/(x − 1). This function is undefined at x = 1 (division by zero).
Factor the numerator: x² − 1 = (x + 1)(x − 1).
Cancel the common factor: (x + 1)(x − 1)/(x − 1) = x + 1, valid for x ≠ 1.
Substitute: lim (x→1) (x + 1) = 2.
The limit is 2, even though f(1) itself is undefined. This is a key point: limits care about the neighbourhood, not the point.
A function f is continuous at x = c when all three of the following hold:
Condition 1: f(c) is defined (the function has an actual output at that point).
Condition 2: lim (x→c) f(x) exists (the left-hand and right-hand limits agree).
Condition 3: lim (x→c) f(x) = f(c) (the limit matches the function value).
If any one of these fails, the function is discontinuous at c.
f(x) = x² is continuous at every real number.
For any c, f(c) = c² is defined, lim (x→c) x² = c², and the limit equals f(c).
There are no holes, jumps, or asymptotes anywhere in the domain.
g(x) = 1/x is not continuous at x = 0.
g(0) is undefined (division by zero), so condition 1 already fails.
From the left, g(x) → −∞. From the right, g(x) → +∞. The one-sided limits disagree, so the two-sided limit does not exist either. Conditions 2 and 3 also fail.
Limit notation: lim (x→c) f(x) = L
Three conditions for continuity at c:
f(c) is defined
lim (x→c) f(x) exists
lim (x→c) f(x) = f(c)
Factoring shortcut used in the example: x² − 1 = (x + 1)(x − 1)
Limits are how engineers and scientists describe rates of change and accumulation, which are the foundations of derivatives and integrals. Any time you hear about velocity at an instant, the slope of a curve, or the area under a graph, limits are doing the heavy lifting behind the scenes. Continuity matters in applied settings because most physical systems (temperature, pressure, position) behave continuously, and that assumption is what makes calculus applicable to the real world.
"If f(c) is undefined, the limit at c cannot exist." Wrong. The limit only cares about what happens near c, not at c. The (x² − 1)/(x − 1) example shows a limit of 2 at x = 1 even though f(1) is undefined.
"If the limit exists, the function is continuous." Not automatically. Continuity also requires that f(c) is defined and that it equals the limit. A function with a hole (removable discontinuity) has a limit but is still discontinuous.
"Infinity is a valid limit value." When we write lim (x→c) f(x) = ∞, we are saying the limit does not exist in the usual sense. The function grows without bound rather than settling on a finite value.
"You can just plug in the value to find any limit." Direct substitution works only when the function is continuous at that point. If substitution gives 0/0 or another indeterminate form, you need algebraic manipulation (factoring, rationalising, etc.) first.
⚠️ The three conditions for continuity are a favourite exam question. You will almost certainly be asked to check each condition for a given function and identify which one fails.
⚠️ Simplifying before substituting (as in the (x² − 1)/(x − 1) example) is a core exam technique. Expect at least one limit that requires factoring or cancellation.
⚠️ One-sided limits: if a problem asks whether a limit exists, check both sides. If they disagree, the answer is that the limit does not exist.
⚠️ Know the difference between "the function is undefined at c" and "the limit at c does not exist." These are separate ideas.
True or false: If f(3) = 7, then lim (x→3) f(x) must equal 7.
A: False. The function value and the limit are only guaranteed to match if f is continuous at that point.
Fill in the blank: For a limit to exist at x = c, the ______ limit and the ______ limit must be equal.
A: left-hand, right-hand.
True or false: The function g(x) = 1/x is continuous on its domain.
A: True. Its domain is all real numbers except 0, and it is continuous at every point in that domain. It is the point x = 0 (which is not in the domain) where the trouble is.
Fill in the blank: lim (x→1) (x² − 1)/(x − 1) = ______.
A: 2.
Q: State the three conditions for a function to be continuous at a point c.
A: (1) f(c) is defined. (2) lim (x→c) f(x) exists. (3) lim (x→c) f(x) = f(c).
Q: Evaluate lim (x→1) (x² − 1)/(x − 1). Show your working.
A: Factor the numerator: (x² − 1) = (x + 1)(x − 1). Cancel (x − 1) with the denominator to get (x + 1). Substitute x = 1: 1 + 1 = 2. The limit is 2.
Q: Is f(x) = (x² − 1)/(x − 1) continuous at x = 1? Explain.
A: No. The function is not defined at x = 1 (division by zero), so condition 1 fails. The limit exists and equals 2, but without a function value there is no continuity.
Q: Explain why g(x) = 1/x is not continuous at x = 0.
A: g(0) is undefined (condition 1 fails). The left-hand limit is −∞ and the right-hand limit is +∞, so the two-sided limit does not exist (condition 2 fails). Since both conditions 1 and 2 fail, g is discontinuous at x = 0.
Q: Give an example of a function where the limit exists at a point but the function is not continuous there.
A: f(x) = (x² − 1)/(x − 1). The limit as x → 1 is 2, but f(1) is undefined, so f is not continuous at x = 1. (Alternatively, any piecewise function where the limit at the boundary exists but the assigned value differs from it.)
This material connects directly to derivatives, which are defined as limits of difference quotients. If you are comfortable with limits, the definition of a derivative will feel like a natural next step rather than a new concept.
Continuity also ties into the Intermediate Value Theorem (IVT), which guarantees that a continuous function on a closed interval takes every value between f(a) and f(b). The IVT is used heavily in root-finding and in proofs throughout analysis.
In later courses, the epsilon-delta definition of a limit formalises the intuition covered here. These notes give you the conceptual foundation; the rigorous definition puts it in proof-ready language.
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