Difficulty: Introductory | Prerequisites: Limits notation and evaluation techniques (see companion notes).
Once you can evaluate limits algebraically, the next question is: what does the limit tell you about the function itself? This topic connects limits to continuity (whether a function has breaks or holes) and teaches you to read limit information directly off a graph. In a business calculus course, this matters because real-world models often have discontinuities (a tax bracket boundary, a pricing tier change), and you need to recognise them. You should already be comfortable with basic limit evaluation before working through this material.
A function is continuous at a point if you can draw through it without lifting your pen: the function is defined there, the limit exists, and the two are equal. When any of those three conditions fails, you have a discontinuity. On a graph, one-sided limits (from the left or right) tell you what the function is approaching, while open and closed circles tell you what the function value actually is.
Continuity at a point
A function f is continuous at x = c if three conditions are all met: (1) f(c) is defined, (2) lim (x→c) f(x) exists, and (3) lim (x→c) f(x) = f(c).
In simple terms, the function arrives where it is supposed to be, with no gap, hole, or jump.
Discontinuity
A point where at least one of the three continuity conditions fails. The function might be undefined there, or the left and right sides might approach different values.
Vertical asymptote (VA)
A vertical line x = c where the function's output grows without bound (heads to +∞ or −∞) as x approaches c. On a graph, the curve shoots up or down near the line and never crosses it.
Think of it as an invisible wall the function cannot pass through.
One-sided limit (left-hand and right-hand)
lim (x→c⁻) f(x) is the value f(x) approaches as x comes from the left (values less than c). lim (x→c⁺) f(x) is the value from the right (values greater than c).
In simple terms, you are checking what happens on each side separately.
Two-sided limit
lim (x→c) f(x) exists only when the left-hand limit and the right-hand limit are equal. If they disagree, the two-sided limit does not exist (DNE).
Open circle (on a graph)
Indicates the function is not defined at that point, or that the function's value is elsewhere. The curve approaches that location but does not include it.
Closed circle (on a graph)
Indicates the function is defined at that point and equals the marked value. The curve includes that point.
f(c) vs lim (x→c) f(x)
f(c) is the actual output of the function at x = c (read from a closed dot). The limit is what the function approaches as x gets near c. These can be different, or one can exist while the other does not.
This section covers what you can say about a function at a particular x-value using limits and algebra.
Worked example: f(x) = (x²+2x−8)/(x²−4) at x = 2
First, factor: f(x) = (x+4)(x−2) / (x−2)(x+2) = (x+4)/(x+2), with the caveat that this simplification is only valid when x ≠ 2.
Two properties at x = 2:
Property 1: f(2) is undefined. The original function has (x−2) in the denominator, so plugging in x = 2 gives 0/0. The function has no output at x = 2.
Property 2: There is a vertical asymptote or discontinuity at x = 2. Specifically, since the (x−2) factor cancels from both numerator and denominator, this is a removable discontinuity (a hole), not a vertical asymptote. The limit lim (x→2) f(x) = 3/2 exists, but f(2) itself does not.
How to tell a hole from a vertical asymptote
If the problematic factor cancels completely from numerator and denominator, the discontinuity is a hole (removable). The limit exists.
If the factor remains in the denominator after cancelling, the function blows up and you have a vertical asymptote. The limit does not exist (the function heads to ±∞).
Graph-reading questions test whether you understand the difference between the function's value at a point and the limit as you approach that point.
One-sided limits
lim (x→c⁻) f(x): trace the curve from the left side and read where it is heading as you approach x = c.
lim (x→c⁺) f(x): trace the curve from the right side.
Two-sided limits
lim (x→c) f(x) exists only if both one-sided limits agree on the same value. If the curve approaches different heights from the left and right, the limit DNE.
Function value f(c)
Look for a closed dot at x = c. That y-coordinate is f(c).
If there is only an open dot (or no dot at all), f(c) is undefined or DNE.
Worked example from the source (graph with discontinuity at x = 2)
lim (x→−2⁺) f(x) = −5. Approaching x = −2 from the right, the curve heads to y = −5.
lim (x→2) f(x) = DNE. The left-hand and right-hand limits at x = 2 disagree.
lim (x→6⁻) f(x) = 6. Approaching x = 6 from the left, the curve reaches y = 6.
lim (x→0) f(x) = 1. Both sides approach y = 1.
f(2) = DNE. The function is not defined at x = 2 (open circle or no value marked).
Worked example from the source (piecewise-style graph)
f(−1) = 2. There is a closed dot at (−1, 2).
lim (x→−1⁻) f(x) = 0. From the left, the curve approaches y = 0.
lim (x→−1⁺) f(x) = 1. From the right, the curve approaches y = 1.
lim (x→−1) f(x) = DNE. The one-sided limits disagree (0 ≠ 1).
f(0) = 1. Closed dot at (0, 1).
lim (x→0⁻) f(x) = 0. From the left, the curve approaches y = 0.
lim (x→0⁺) f(x) = 0. From the right, the curve also approaches y = 0.
lim (x→0) f(x) = 0. Both sides agree, so the two-sided limit is 0. Note: f(0) = 1, which is different from the limit. The function "jumps" at x = 0.
lim (x→1) f(x) = DNE. The one-sided limits at x = 1 disagree.
Discontinuities appear in business models more often than you might expect. A stepped pricing function (one price for the first 100 units, a lower price after that) is discontinuous at the boundary. Tax brackets create jumps in effective tax rate functions. Recognising these on a graph, and knowing when a limit exists even though the function value does not, is the mathematical foundation for analysing such models.
Students often confuse f(c) with lim (x→c) f(x). They are different questions. f(c) asks "what is the output at c?" The limit asks "what is the output approaching near c?" They can disagree.
Students sometimes assume that if a function is undefined at a point, the limit also does not exist there. That is wrong. The classic example: (x²+2x−8)/(x²−4) is undefined at x = 2, but the limit at x = 2 is 3/2.
When reading graphs, students mix up open and closed circles. An open circle means the point is excluded; a closed circle means it is included. Getting this backwards changes your answer for f(c).
Students sometimes forget to check both sides before declaring a two-sided limit. If the left-hand and right-hand limits differ, the two-sided limit DNE, regardless of whether the function happens to be defined there.
⚠️ Graph-reading questions are on nearly every exam. You will be given a graph and asked for one-sided limits, two-sided limits, and function values. Practise reading all three from the same graph.
⚠️ "State two properties of f(x) at x = c" is a common exam prompt. Know how to identify whether f(c) is defined, whether the limit exists, and whether there is a hole or a vertical asymptote.
⚠️ The distinction between "limit exists but function is undefined" (a hole) and "function blows up" (a vertical asymptote) is frequently tested.
⚠️ DNE questions: be precise about when to write DNE. Use it for two-sided limits where the one-sided limits disagree, and for f(c) when the function is undefined at c.
True or False: If f(c) is undefined, the limit at x = c must also not exist.
False. The limit can exist even when the function is undefined at that point (a removable discontinuity).
Fill in the blank: A two-sided limit exists only when the ________ limit and the ________ limit are equal.
Left-hand, right-hand.
True or False: An open circle on a graph means the function equals that y-value at that x.
False. An open circle means the function does not include that point. A closed circle means it does.
Fill in the blank: If the (x−c) factor cancels from both numerator and denominator of a rational function, the discontinuity at x = c is a ________, not a vertical asymptote.
Hole (removable discontinuity).
True or False: If a graph shows the curve approaching y = 3 from the left and y = 5 from the right at x = 2, then lim (x→2) f(x) = 3.
False. The two-sided limit does not exist because the one-sided limits disagree.
Q: Given f(x) = (x²+2x−8)/(x²−4), state two properties of f at x = 2.
A: (1) f(2) is undefined because the denominator equals zero at x = 2. (2) There is a removable discontinuity (hole) at x = 2 because the (x−2) factor cancels, and the limit lim (x→2) f(x) = 3/2 exists.
Q: From a graph, lim (x→−2⁺) f(x) = −5 and lim (x→−2⁻) f(x) = 10. What is lim (x→−2) f(x)?
A: DNE. The left-hand limit (10) and the right-hand limit (−5) are not equal, so the two-sided limit does not exist.
Q: A graph shows an open circle at (0, 1) and the curve approaching y = 0 from both sides. What is f(0)? What is lim (x→0) f(x)?
A: f(0) is read from the closed dot. If there is only an open circle at y = 1 and a closed dot at, say, y = 1, then f(0) = 1. The limit lim (x→0) f(x) = 0, because the curve approaches 0 from both sides. Note: the limit and the function value are different.
Q: How do you distinguish a hole from a vertical asymptote in a rational function?
A: Factor the function. If the factor causing 0/0 cancels completely from both numerator and denominator, the discontinuity is a hole and the limit exists. If the factor remains in the denominator after cancelling, the function approaches ±∞ and there is a vertical asymptote.
Q: A graph shows f(−1) = 2 (closed dot), lim (x→−1⁻) f(x) = 0, and lim (x→−1⁺) f(x) = 1. Does lim (x→−1) f(x) exist?
A: No. The one-sided limits (0 and 1) are not equal, so the two-sided limit DNE. The function value f(−1) = 2 is irrelevant to whether the limit exists.
Continuity is a prerequisite for the derivative. A function must be continuous at a point for the derivative to exist there, so the continuity checks you learn here come back in every differentiation chapter.
Graph-reading skills also connect to optimisation problems later in the course, where you need to identify where a function changes direction, and to the Intermediate Value Theorem, which relies on continuity to guarantee that a function hits every value between its endpoints.
Continuity, discontinuity, removable discontinuity, hole in a graph, vertical asymptote, one-sided limit, left-hand limit, right-hand limit, two-sided limit, DNE, does not exist, open circle, closed circle, piecewise function, graph reading, function value vs limit, f(c) vs limit, business calculus, calculus for business, FIU calculus, polynomial functions, rational functions, continuity conditions