Difficulty: Introductory | Prerequisites: Basic algebra, function notation (f(x), y = f(x)), familiarity with the Cartesian plane.
This is foundational material for any calculus course, and it sits right at the beginning for good reason. Limits describe what a function does as its input gets close to some value, and that idea underpins everything that follows: derivatives, integrals, continuity. If you are coming from a pre-calculus or college algebra background, this is where calculus properly starts. You need to be comfortable reading graphs and understanding x-y coordinates before this will make sense.
A limit asks: "What value does y approach as x gets close to some number (or heads off to infinity)?" You can approach from the left side only, the right side only, or both. If the left-hand and right-hand limits disagree, or if the function blows up to infinity, the two-sided limit Does Not Exist (DNE).
Limit
The value that a function's output (y) approaches as its input (x) gets arbitrarily close to a particular number. In simple terms, it is the answer to "where is y headed?" as x closes in on some target.
One-sided limit (left-hand limit)
Written lim x→c⁻. The value y approaches as x comes in from values less than c (from the left on the number line). Think of it as sneaking up on c from the smaller side only.
One-sided limit (right-hand limit)
Written lim x→c⁺. The value y approaches as x comes in from values greater than c (from the right on the number line). Think of it as sneaking up on c from the larger side only.
Two-sided limit
Written lim x→c. This limit exists only when the left-hand and right-hand limits are equal. In simple terms, both directions have to agree, or the two-sided limit does not exist.
Does Not Exist (DNE)
A limit is said to not exist when the function does not approach a single finite value. This happens when the left and right sides approach different values, or when the function heads to positive or negative infinity, or when the function oscillates without settling.
Unbounded behaviour / "increases (or decreases) without bound"
The function's output grows towards +∞ or −∞ rather than levelling off at a finite number. In simple terms, y keeps getting bigger (or more negative) forever, with no ceiling or floor.
Vertical asymptote
A vertical line x = c where the function's output blows up to ±∞. Limits at a vertical asymptote are always DNE in the two-sided sense, though the one-sided limits may each be +∞ or −∞.
To find a limit from a graph, trace the curve towards the target x-value and read off where y is heading.
You are not asking "what is y at that point?" You are asking "what value is y approaching?"
The function does not need to actually reach the limit value. It might have a hole, a jump, or an asymptote at that x-value. The limit only cares about the approach.
Left-hand limit, lim x→c⁻: cover the right side of the graph and look only at what y does as x approaches c from below (from the left).
Right-hand limit, lim x→c⁺: cover the left side of the graph and look only at what y does as x approaches c from above (from the right).
Example from the practice set: at x = −3, the left-hand limit gives y = 3, while the right-hand limit gives y → +∞ (DNE). Because they disagree, the two-sided limit at x = −3 is DNE.
The two-sided limit lim x→c exists only when lim x→c⁻ = lim x→c⁺ (both sides approach the same finite value).
Example from the practice set: at x = 0, y approaches −1 from both sides. So lim x→0 y = −1.
When y grows without bound (heads to +∞ or −∞) as x approaches some value, the limit is DNE. We write lim = +∞ or lim = −∞ to describe the behaviour, but the limit does not exist as a finite number.
Example from the practice set: at x = −2, y → +∞ from the left and y → −∞ from the right. Both one-sided limits are unbounded, and the two-sided limit is DNE.
These ask: what happens to y when x itself heads off to positive or negative infinity?
If y settles towards a horizontal value, that value is the limit. This often corresponds to a horizontal asymptote.
Example from the practice set: as x → −∞, y decreases without bound (so the limit is −∞, DNE). As x → +∞, y approaches 1, so lim x→+∞ y = 1.
Your course expects you to translate limit notation into plain English. The pattern is straightforward:
"As x approaches [value] from [direction], y approaches [value]." If unbounded: "As x approaches [value] from [direction], y decreases (or increases) without bound."
Example: lim x→−3⁻ y = 3 becomes "As x approaches −3 from the left, y approaches 3."
Example: lim x→−2⁻ y = +∞ becomes "As x approaches −2 from the left side, y decreases without bound." (Note: check the direction of unboundedness carefully from the graph.)
\lim_{x \to c} f(x) = LTwo-sided limit: y approaches L as x approaches c from both directions.
\lim_{x \to c^{-}} f(x) = LLeft-hand limit: y approaches L as x approaches c from values less than c.
\lim_{x \to c^{+}} f(x) = LRight-hand limit: y approaches L as x approaches c from values greater than c.
The two-sided limit exists if and only if:
\lim_{x \to c^{-}} f(x) = \lim_{x \to c^{+}} f(x) = LIf the one-sided limits are not equal, or if either is ±∞, the two-sided limit is DNE.
In business calculus, limits show up whenever you need to understand behaviour at a boundary. Marginal cost (the cost of producing one more unit) is defined using limits. Break-even analysis relies on understanding what happens to profit as quantity approaches a critical threshold. In finance, compound interest taken to its logical extreme (continuous compounding) is literally a limit: what happens as the number of compounding periods grows without bound.
Students often confuse the value of f(c) with the limit as x→c. The function can equal 5 at x = c while the limit is 3 (a removable discontinuity). The limit is about the approach, not the arrival.
Students sometimes write "lim = ∞" and treat that as the limit existing. If the function heads to infinity, the limit Does Not Exist. Writing "= ∞" is a description of the behaviour, not a finite answer.
Forgetting to check both sides is a common error. A limit from the left might be perfectly well-behaved while the limit from the right blows up. You must verify both before declaring the two-sided limit.
Mixing up the superscript notation: the minus sign in x→c⁻ means "from the left" (from smaller values), and the plus sign in x→c⁺ means "from the right" (from larger values). This trips up students who associate minus with right and plus with left on a number line.
⚠️ Expect graph-based limit questions. The practice set is entirely graphical, so you should be able to read one-sided and two-sided limits off a curve without any algebra.
⚠️ Sentence-form answers are explicitly tested. Your course requires you to write out the limit in words ("As x approaches..."), not just state the numerical answer.
⚠️ DNE is a valid answer and comes up frequently. Know the three main reasons a limit is DNE: left and right limits disagree, the function is unbounded, or the function oscillates.
⚠️ Be precise about direction when a function is unbounded. "y increases without bound" and "y decreases without bound" are different answers, and the exam will distinguish them.
True or False: If lim x→c⁻ f(x) = 4 and lim x→c⁺ f(x) = 4, then lim x→c f(x) = 4. (True.)
True or False: If lim x→c f(x) = ∞, then the limit exists. (False. The limit is DNE.)
Fill in the blank: The notation x→c⁻ means x approaches c from the ___. (Left, i.e. from values less than c.)
True or False: A function must be defined at x = c for the limit as x→c to exist. (False. The limit depends on the approach, not the value at the point.)
Fill in the blank: If the left-hand limit is 2 and the right-hand limit is 5, the two-sided limit is ___. (DNE.)
Q: A graph shows y heading toward 3 as x approaches 5 from the left, and y heading toward 7 as x approaches 5 from the right. What is lim x→5 f(x)? Write your answer in sentence form.
A: The limit does not exist. As x approaches 5 from the left, y approaches 3, and as x approaches 5 from the right, y approaches 7. Because the one-sided limits are not equal, the two-sided limit is DNE.
Q: On a graph, as x approaches −4 from both sides, y gets larger and larger without stopping. What is lim x→−4 f(x)?
A: The limit does not exist. As x approaches −4 from both sides, y increases without bound. We write lim x→−4 f(x) = +∞ to describe the behaviour, but the limit is DNE because it is not a finite value.
Q: Explain the difference between lim x→2⁻ f(x) = 6 and f(2) = 6.
A: The first statement says that as x approaches 2 from the left, y approaches 6. The second says the function's actual output at x = 2 is 6. These can be true independently. The limit is about the trend of nearby values; f(2) is the value at the point itself. A function can have a limit of 6 as x→2 while f(2) is undefined or equals something else entirely.
Q: As x → +∞, a function levels off and gets closer and closer to y = 4. What is lim x→+∞ f(x)? Write your answer in sentence form.
A: As x increases without bound, y approaches 4. So lim x→+∞ f(x) = 4. The line y = 4 is a horizontal asymptote of the function.
Q: A graph has a vertical asymptote at x = 1. From the left, y → −∞. From the right, y → +∞. State the one-sided limits and the two-sided limit.
A: lim x→1⁻ f(x) = −∞ (DNE), lim x→1⁺ f(x) = +∞ (DNE), and lim x→1 f(x) = DNE. In sentence form: as x approaches 1 from the left, y decreases without bound; as x approaches 1 from the right, y increases without bound. The two-sided limit does not exist because the one-sided behaviours differ.
Limits feed directly into continuity: a function is continuous at x = c precisely when lim x→c f(x) = f(c). Once you have limits down, the definition of a derivative is next, because the derivative is itself a limit (the limit of the difference quotient as the interval shrinks to zero). In business calculus specifically, marginal analysis (marginal cost, marginal revenue, marginal profit) is built on derivatives, which means it is built on limits.
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