Difficulty: Introductory | Prerequisites: Algebra fundamentals, function notation
Limits are the foundational idea underpinning all of calculus. Before you can understand derivatives or integrals, you need to understand what it means for a function to approach a value. This chapter introduces the notation, the rules for combining limits, and the techniques for evaluating them, including what to do when direct substitution gives you something undefined. If you are comfortable with basic algebra and function notation, you are ready for this material.
A limit describes the value a function approaches as x gets close to some number, even if the function never reaches that value. You can evaluate limits by direct substitution, factoring, using conjugates, or applying limit operation rules. When direct substitution gives 0/0, there is more work to do (factor, cancel, or rationalise).
Limit
The value that a function f(x) approaches as x approaches a particular number. Written as lim (x→a) f(x) = L. In simple terms, it is the y-value the graph is heading towards as you get closer and closer to a given x-value, even if the function never lands on it.
Two-sided limit
The overall limit of f(x) as x→a. It exists only when the left-hand limit and the right-hand limit both exist and are equal. Think of it as: both directions have to agree, or there is no limit.
One-sided limit (left-hand and right-hand)
The value f(x) approaches from one direction only. The left-hand limit (x→a⁻) looks at x-values less than a. The right-hand limit (x→a⁺) looks at x-values greater than a. In simple terms, you are sneaking up on a from just one side.
Vertical asymptote
A vertical line x = a where the function grows without bound (heads to +∞ or −∞). After plugging in x = a by direct substitution, if the result is some nonzero number divided by 0, there is a vertical asymptote.
Hole (removable discontinuity)
A point where the function is undefined (or defined differently) but the limit still exists. Occurs when direct substitution gives 0/0, which signals a common factor you can cancel. The limit equals the reduced function's value when a is plugged in, and the graph shows an open circle at that point.
Table of values
A method for estimating limits by choosing x-values increasingly close to a from both sides and observing whether the outputs converge. Think of it as: plug in numbers near a and watch where f(x) is heading.
Direct substitution
The simplest limit technique: plug a straight into f(x). If you get a finite number, that is the limit. If you get 0/0 (indeterminate), further algebraic work is needed. If you get a nonzero number over 0, there is a vertical asymptote.
The limit of f(x) as x→a is the y-value the function is heading towards, not necessarily the value it reaches at x = a.
A two-sided limit exists if and only if the left-hand limit and the right-hand limit both exist and are equal.
If the left and right sides disagree, the limit Does Not Exist (DNE).
Plug a into f(x).
If you get a real number, that is the limit.
If you get a nonzero number over 0 (e.g. 5/0), there is a vertical asymptote.
If you get 0/0, the result is indeterminate and you need to simplify further.
A result of 0/0 means there is a common factor in the numerator and denominator.
Factor and cancel that common factor (or use the conjugate technique for radicals).
Plug a into the reduced function to find the limit.
On the graph, the original function has an open circle (hole) at that x-value, and the reduced function fills the gap.
Choose x-values approaching a from the left (e.g. 1.9, 1.99, 1.999) and from the right (e.g. 2.1, 2.01, 2.001).
Compute f(x) for each.
If both sides converge to the same output, that output is the limit.
Useful for confirming an algebraic answer or when the algebra is not straightforward.
Trace the graph from the left towards x = a: the y-value you approach is the left-hand limit.
Trace the graph from the right towards x = a: the y-value you approach is the right-hand limit.
If both sides arrive at the same y-value, the two-sided limit exists and equals that value.
An open circle on the graph indicates the function is not defined at that point, but the limit can still exist.
Given lim (x→a) f(x) = L₁ and lim (x→a) g(x) = L₂:
Sum: lim [f(x) + g(x)] = L₁ + L₂
Difference: lim [f(x) − g(x)] = L₁ − L₂
Product: lim [f(x) · g(x)] = L₁ · L₂
Quotient: lim [f(x) / g(x)] = L₁ / L₂, provided L₂ ≠ 0
Power/Root: lim [f(x)]^(1/n) = (L₁)^(1/n), provided L₁ ≥ 0 when n is even
All of these rules also hold for one-sided limits.
End behaviour describes what f(x) does as x→+∞ or x→−∞.
For a polynomial, end behaviour is determined entirely by its leading term.
For a rational function (polynomial over polynomial):
Degree of numerator < degree of denominator: horizontal asymptote at y = 0.
Degree of numerator = degree of denominator: horizontal asymptote at y = ratio of leading coefficients.
Degree of numerator > degree of denominator: no horizontal asymptote (the function grows without bound).
The y-value of the horizontal asymptote is the limit as x→±∞.
Limit operation rules (assuming both individual limits exist):
\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{provided } \lim_{x \to a} g(x) \neq 0Horizontal asymptote decision rule for rational functions p(x)/q(x):
deg(p) < deg(q): limit as x→±∞ is 0
deg(p) = deg(q): limit as x→±∞ is the ratio of leading coefficients
deg(p) > deg(q): limit as x→±∞ is ±∞ (no horizontal asymptote)
Students often think the limit of f(x) at x = a must equal f(a). It does not. The limit is about where the function is heading, not where it is. A function can have a hole at x = 2 but still have a perfectly well-defined limit there.
Students often confuse 0/0 with "the limit does not exist." A result of 0/0 is indeterminate, meaning more work is needed. It usually signals a hole, and the limit almost certainly does exist once you simplify.
Students often assume that a vertical asymptote means the limit is infinity. The left-hand side might go to +∞ while the right-hand side goes to −∞ (or vice versa), in which case the two-sided limit does not exist.
Students sometimes apply the quotient rule for limits even when the denominator's limit is 0. The quotient rule requires L₂ ≠ 0. When the denominator's limit is 0, you need a different approach entirely.
⚠️ Direct substitution is always the first step. If you skip it and jump straight to factoring, you may do unnecessary work or miss that the answer is simply f(a).
⚠️ Know the difference between a hole and a vertical asymptote. Exams love giving you a rational function and asking you to identify which x-values produce which type of discontinuity.
⚠️ Horizontal asymptote problems are a staple. Be fluent in the three-case rule for rational functions (compare degrees of numerator and denominator).
⚠️ When reading a graph, pay close attention to open vs closed circles. An open circle at a point means the function is not defined there (or is defined elsewhere), but the limit can still exist.
True or False: If f(3) = 7, then lim (x→3) f(x) must also equal 7.
Fill in the blank: When direct substitution gives 0/0, the result is called __________ and requires further simplification.
True or False: A two-sided limit exists even when the left-hand and right-hand limits are different values.
Fill in the blank: For a rational function where the degree of the numerator equals the degree of the denominator, the horizontal asymptote is y = __________.
True or False: A vertical asymptote occurs when direct substitution gives a nonzero number divided by 0.
Answers: 1. False (the limit depends on behaviour near 3, not at 3). 2. Indeterminate. 3. False (they must be equal). 4. The ratio of the leading coefficients. 5. True.
Q: Evaluate lim (x→4) (x² − 16) / (x − 4).
A: Direct substitution gives 0/0. Factor the numerator: (x − 4)(x + 4) / (x − 4). Cancel the common factor to get x + 4. Plug in x = 4: the limit is 8.
Q: What is lim (x→∞) (3x² + 5) / (7x² − 2)?
A: The degrees of numerator and denominator are equal (both degree 2). The limit is the ratio of leading coefficients: 3/7.
Q: A function has a vertical asymptote at x = 1. What did direct substitution produce at x = 1?
A: A nonzero number divided by 0 (e.g. 5/0 or −3/0). This signals the function blows up near x = 1.
Q: From a graph, the left-hand limit as x→2 is 5 and the right-hand limit as x→2 is 3. Does lim (x→2) f(x) exist?
A: No. The two-sided limit exists only when both one-sided limits are equal. Since 5 ≠ 3, the limit does not exist.
Q: Evaluate lim (x→0) [4f(x) − g(x)] given that lim (x→0) f(x) = 3 and lim (x→0) g(x) = −1.
A: Apply the limit operation rules: 4(3) − (−1) = 12 + 1 = 13.
Limits are the formal machinery behind derivatives. In Ch. 2, the derivative is defined as a limit of a difference quotient, so every technique you learn here (direct substitution, factoring, cancelling) will reappear in that context.
Continuity (covered in Sections 1.5 and 1.6) is defined in terms of limits. A function is continuous at a point precisely when the limit equals the function value, which ties these two ideas together directly.
Horizontal asymptotes connect to end behaviour of rational functions, which matters again in curve sketching and optimisation problems later in the course.
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