Difficulty: Introductory | Prerequisites: Basic function notation, reading graphs on the coordinate plane.
This is typically the first topic in a calculus course. Before derivatives and integrals enter the picture, you need to understand what a function is doing as its input creeps toward a particular value. Limits from graphs are the visual, intuitive version of that idea, and they lay the groundwork for continuity, derivatives, and everything that follows. If you missed the first week, start here.
A limit asks: "What y-value is the graph heading toward as x gets close to some number?" You can approach from the left side only, the right side only, or both. If the left-hand and right-hand limits agree, the overall (two-sided) limit exists and equals that shared value. If they disagree, the limit does not exist (DNE).
Limit of f(x) as x approaches a
The value that f(x) gets arbitrarily close to as x gets close to a, written lim x→a f(x) = L. Think of it as: "Where is the graph heading when x is near a?" You are not asking what happens at x = a, only what happens near it.
Left-hand limit (limit from the left)
The value f(x) approaches as x comes from values less than a, written lim x→a⁻ f(x). In simple terms, this means you trace the graph moving rightward toward x = a, and read off the y-value the curve is heading toward.
Right-hand limit (limit from the right)
The value f(x) approaches as x comes from values greater than a, written lim x→a⁺ f(x). In simple terms, this means you trace the graph moving leftward toward x = a, and read off the y-value the curve is heading toward.
DNE (Does Not Exist)
A limit "does not exist" when there is no single finite value the function approaches. The most common reason at this level: the left-hand and right-hand limits give different y-values. Written as: lim x→a f(x) = DNE.
Open circle (open dot) on a graph
A hollow circle at a point means the function is not defined there, or is defined at a different value. The graph approaches that point but does not include it. Think of it as a hole in the graph.
Closed circle (filled dot) on a graph
A solid dot at a point means the function is defined there and equals that y-value. This is the actual value of f(a). Think of it as the function's "official answer" at that x-value.
Step function
A function whose graph looks like a staircase, jumping between constant values at certain x-values. Taxi fare functions are a classic example: the cost stays flat, then jumps at each distance threshold.
A two-sided limit asks what y-value the graph is heading toward as x approaches a from both sides simultaneously. If you can put your finger on the graph to the left of x = a and slide it rightward, and separately put your finger to the right of x = a and slide it leftward, and both fingers end up at the same height, that height is the limit.
How to read it: Draw a mental vertical line at x = a. Follow the curve in from both directions. Read the y-value where the curve is heading, not where a dot might be plotted.
The actual value of f(a) does not matter. The function might have a hole, a dot plotted elsewhere, or not be defined at all. The limit only cares about the trend.
Worked examples from the source material:
lim x→3 f(x) = 8. The graph heads toward y = 8 from both sides as x approaches 3, so the limit is 8.
lim x→7 f(x) = 1. Even though there is an open circle at (7, 1), both sides of the graph head toward y = 1. The open circle means f(7) ≠ 1 (or f(7) is undefined), but the limit is still 1.
lim x→4 H(x) = 15. Both sides of H(x) head toward 15 as x approaches 4.
lim x→−2 H(x) = 3. Both sides agree on y = 3, despite the function value at x = −2 possibly being different (an open circle sits at the point the graph heads toward).
lim x→1 f(x) = −3. The graph of f heads toward y = −3 from both sides as x → 1.
lim x→−3 H(x) = 1. Both sides of H(x) converge on y = 1 near x = −3.
Sometimes your exam asks for the limit from one direction only. The superscript minus or plus sign tells you which way.
lim x→a⁻ f(x) means approach a from the left (from smaller x-values). On the graph, slide your finger along the curve from the left side toward the vertical line x = a. The y-value your finger reaches is the left-hand limit.
lim x→a⁺ f(x) means approach a from the right (from larger x-values). Slide your finger from the right side toward x = a. The y-value your finger reaches is the right-hand limit.
Key technique: The source material describes this as finding the "first line it hits." Imagine drawing the vertical line x = a. As you trace the curve from the specified direction, the first y-value the curve meets at that vertical line is your answer.
Worked examples from the source material:
lim x→−5⁺ f(x) = 3. Approaching x = −5 from the right, the graph heads toward y = 3.
lim x→3⁻ G(x) = 7. Approaching x = 3 from the left, the graph heads toward y = 7.
lim x→2⁻ G(x): Approaching x = 2 from the left, trace the curve and read the y-value where it arrives at the vertical line x = 2.
lim x→1⁻ H(x) = 4. From the left of x = 1, the graph heads toward y = 4. The note says "touches this line first," meaning the curve reaches y = 4 as it arrives at x = 1.
lim x→3⁻ G(x) = 6. From the left of x = 3, the graph heads toward y = 6.
A two-sided limit does not exist when the left-hand and right-hand limits disagree. At this level of calculus, the most common DNE scenario is a jump discontinuity: the graph approaches one y-value from the left and a different y-value from the right.
The test: Compute lim x→a⁻ f(x) and lim x→a⁺ f(x) separately. If these two values are not equal, write lim x→a f(x) = DNE.
Worked examples from the source material:
lim x→−1 f(x) = DNE. From the left the graph heads toward y = 5, from the right it heads toward y = 2. Since 5 ≠ 2, the limit does not exist.
lim x→3 G(x) = DNE. From the left the graph heads toward y = 5, from the right it heads toward y = −2. Since 5 ≠ −2, the limit does not exist.
lim x→1 H(x) = DNE. From the left the graph heads toward y = 6, from the right it heads toward y = 0. Since 6 ≠ 0, the limit does not exist.
lim x→1 H(x) = DNE (another graph). From the left the graph heads toward y = 5, from the right toward y = −2. Since 5 ≠ −2, DNE.
Important: When writing DNE on an exam, briefly show your working. Write both one-sided limits and state that they are not equal. This is usually worth partial marks even if you make an error reading one of the values.
The fundamental rule for limit existence:
lim x→a f(x) = L if and only if lim x→a⁻ f(x) = L and lim x→a⁺ f(x) = L.
In words: a two-sided limit equals L only when both one-sided limits exist and both equal L. If the one-sided limits give different numbers, or if either one fails to settle on a finite value, the two-sided limit does not exist.
Notation summary:
lim x→a f(x) = two-sided limit (approaching from both directions)
lim x→a⁻ f(x) = left-hand limit (x approaches a from below, i.e. x < a)
lim x→a⁺ f(x) = right-hand limit (x approaches a from above, i.e. x > a)
DNE = "does not exist"
Students often think the limit equals f(a). It does not, necessarily. The limit is about where the graph is heading, not where a dot is plotted. A function can have f(3) = 10 while lim x→3 f(x) = 8, if the graph heads toward 8 but has a point plotted at (3, 10).
Students confuse open and closed dots with the limit value. Open and closed dots tell you the actual value of the function (or the absence of one). They do not change the limit. The limit reads the trend of the curve, not the dot.
Students forget to check both sides before declaring a two-sided limit. If the problem says lim x→a f(x) (no superscript), you must verify left and right agree. Writing down the y-value from one direction alone is not sufficient.
Students write DNE when they mean "the function is undefined." These are different ideas. f(a) can be undefined (there is a hole) while lim x→a f(x) exists perfectly well, if both sides agree. DNE is reserved for the limit itself, not for the function value.
⚠️ Limits from graphs are nearly always on early exams. You will be given a graph and asked to evaluate several limits (two-sided, left, right) at different x-values. Speed and accuracy in reading graphs matters.
⚠️ The "trick" question: a graph with a hole at (a, L) and a filled dot at (a, M). The limit is L (the trend), not M (the plotted value). Expect at least one of these.
⚠️ When the problem asks for a two-sided limit and the graph has a jump, show both one-sided limits and state they are unequal. Writing only "DNE" without justification may lose marks.
⚠️ One-sided limit notation matters. The minus sign in x→a⁻ means "from the left" (smaller values), and the plus in x→a⁺ means "from the right" (larger values). Mixing these up is an easy, avoidable error.
True or False: If f(3) = 5, then lim x→3 f(x) must also equal 5. Answer: False. The function value and the limit are independent. The limit depends on the trend of the graph near x = 3, not the value at x = 3.
Fill in the blank: lim x→a f(x) exists if and only if lim x→a⁻ f(x) = lim x→a⁺ f(x) = ____. Answer: L (some common finite value).
True or False: An open circle at a point on the graph means the limit at that x-value does not exist. Answer: False. An open circle means the function is not defined there (or is defined elsewhere). The limit can still exist.
Fill in the blank: The superscript ⁻ in lim x→4⁻ f(x) means approach 4 from the ____. Answer: left (from values less than 4).
True or False: If the left-hand limit is 7 and the right-hand limit is 7, the two-sided limit is 7 regardless of whether f(a) is defined. Answer: True.
Q: A graph of f(x) shows the curve heading toward y = 5 from both sides as x → 2, but there is an open circle at (2, 5) and a closed dot at (2, 1). What is lim x→2 f(x)?
A: The limit is 5. The curve heads toward y = 5 from both sides, which is all the limit cares about. The closed dot at (2, 1) tells you f(2) = 1, but that does not affect the limit.
Q: On a graph, lim x→4⁻ g(x) = 3 and lim x→4⁺ g(x) = 7. What is lim x→4 g(x)?
A: DNE. The left-hand limit (3) does not equal the right-hand limit (7), so the two-sided limit does not exist.
Q: The graph of H(x) approaches y = 6 from both sides of x = −1, and H(−1) = 6 (closed dot). What is lim x→−1 H(x)?
A: The limit is 6. Both sides agree, so the limit exists. The fact that H(−1) also equals 6 is a bonus (it means H is continuous there), but the limit would be 6 regardless of the function value.
Q: You are asked to find lim x→5⁻ f(x). In which direction do you trace the graph?
A: From the left. You start at x-values less than 5 and move rightward toward x = 5, reading the y-value the curve approaches.
Q: A graph shows f(x) heading toward y = 4 as x → 3 from both sides. There is no dot at x = 3 at all (f is undefined there). Does the limit exist?
A: Yes. lim x→3 f(x) = 4. The function does not need to be defined at a point for the limit to exist there.
This connects to continuity because a function is continuous at x = a only when three things hold: f(a) is defined, lim x→a f(x) exists, and f(a) equals the limit. Every continuity problem starts with a limit check.
This connects to derivatives because the derivative at a point is itself defined as a limit (the limit of the difference quotient). If you cannot evaluate limits confidently, derivatives will not make sense.
This connects to piecewise and step functions (covered in Part 2 of these notes) because those are the graphs most likely to produce one-sided limits and DNE results.
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