What you'll learn
Limits are the foundation of calculus — the idea of what a function approaches lets us define derivatives and integrals rigorously. This is the focus of Limits and Continuity (Unit 1 of AP Calculus AB). For AP Calculus AB you need to understand what a limit is, how to evaluate limits graphically, numerically and algebraically, one-sided limits, limits involving infinity, and the definition of continuity. This guide covers the meaning of a limit, techniques for evaluating limits, continuity and its types of discontinuity, and key theorems. By the end you should be able to evaluate limits and determine whether a function is continuous.
Key terms and definitions
Limit — The value a function approaches as the input approaches a particular value.
One-sided limit — The value a function approaches from only the left or only the right.
Continuity — A function is continuous at a point if there is no break, jump or hole there.
Discontinuity — A point where a function is not continuous.
Removable discontinuity — A "hole" that could be filled to make the function continuous.
Asymptote — A line the graph approaches but does not touch.
Indeterminate form — An expression like 0/0 that needs further work to evaluate.
Intermediate Value Theorem — A theorem guaranteeing a value is taken between two others for a continuous function.
Core concepts
What a limit is
The limit of a function f(x) as x approaches a value c is the value that f(x) gets close to as x gets close to c. It is written as lim(x→c) f(x). Importantly, the limit describes what the function approaches, which may be different from the function's actual value at c — or the function may not even be defined at c, yet the limit can still exist.
Evaluating limits graphically and numerically
You can find a limit by looking at a graph — trace the curve towards x = c from both sides and see what y-value it approaches. You can also find it numerically, by making a table of f(x) for x values getting closer and closer to c from both sides, and seeing what value the outputs approach. These methods build intuition, but algebraic methods are usually needed for exact answers.
Evaluating limits algebraically
The main algebraic techniques are:
- Direct substitution — substitute x = c; if you get a defined number, that is the limit.
- Factoring and cancelling — if substitution gives the indeterminate form 0/0, factor the numerator and denominator, cancel the common factor, then substitute.
- Rationalising — for expressions with square roots, multiply by the conjugate to simplify.
- Simplifying complex fractions — combine fractions before substituting.
Direct substitution is always the first thing to try; the other methods are used when it gives 0/0.
One-sided limits
A one-sided limit considers the function approaching c from only one side:
- The left-hand limit (x → c⁻) considers values less than c.
- The right-hand limit (x → c⁺) considers values greater than c.
The two-sided limit exists only if the left-hand and right-hand limits are equal. If they differ, the limit does not exist. One-sided limits are especially important for piecewise functions and points of discontinuity.
Limits involving infinity
Limits can involve infinity in two ways:
- Limits at infinity (x → ∞ or x → −∞) describe the end behaviour of a function, and relate to horizontal asymptotes. For rational functions, compare the degrees of the numerator and denominator to find the limit.
- Infinite limits (the function grows without bound as x → c) relate to vertical asymptotes.
Understanding these connects limits to the shape and asymptotes of a graph.
Continuity
A function is continuous at x = c if three conditions all hold:
- f(c) is defined (the function has a value there).
- The limit lim(x→c) f(x) exists (left and right limits are equal).
- The limit equals the function value: lim(x→c) f(x) = f(c).
If any condition fails, the function is discontinuous at c. Types of discontinuity include removable (a hole, where the limit exists but does not equal f(c) or f(c) is undefined), jump (the one-sided limits differ), and infinite (a vertical asymptote).
Key theorems
Two theorems are important in this unit. The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b) at least once — useful for showing a solution exists. The Squeeze Theorem finds a limit by trapping a function between two others that have the same limit. Both rely on continuity and limits.
Why limits matter for the rest of calculus
It is worth understanding why this unit comes first. The two central ideas of calculus — the derivative and the definite integral — are both defined using limits. The derivative, which measures the rate of change of a function, is defined as the limit of the average rate of change (a difference quotient) as the interval shrinks to zero. The definite integral, which measures accumulated area, is defined as the limit of a sum of rectangles as their width approaches zero. So every later topic rests on the idea of a limit. Mastering limits now means the definitions of the derivative and integral will make sense later, rather than appearing as formulas to memorise. This is why examiners test limits so thoroughly at the start of the course.
Removable discontinuities and simplifying
A common situation is a removable discontinuity — a single "hole" in an otherwise continuous graph. This happens when a factor cancels from a rational function, such as (x² − 4)/(x − 2), which simplifies to x + 2 everywhere except at x = 2, where the original function is undefined. The limit at x = 2 exists (it is 4), even though the function has no value there, so the discontinuity is removable: you could "fill the hole" by defining the function to equal 4 at that point. Recognising that a cancelled factor produces a removable discontinuity, and being able to state both the limit and why the function is discontinuous there, connects the algebra of limits to the idea of continuity and is frequently tested.
Worked examples
Example 1: Direct substitution
Evaluate lim(x→3) (2x + 1). Substituting x = 3 gives 2(3) + 1 = 7. Since substitution gives a defined value, that is the limit.
Example 2: Factoring a 0/0 form
Evaluate lim(x→2) (x² − 4)/(x − 2). Substituting gives 0/0, so factor: (x² − 4) = (x − 2)(x + 2). Cancel (x − 2): the limit becomes lim(x→2) (x + 2) = 4.
Example 3: A one-sided check
A piecewise function gives left-hand limit 5 and right-hand limit 3 at x = 1. Does the limit exist? No. Because the left-hand limit (5) and right-hand limit (3) are not equal, the two-sided limit at x = 1 does not exist.
Example 4: Checking continuity
Is f continuous at x = 2 if f(2) = 4 and lim(x→2) f(x) = 4? Yes. f(2) is defined (4), the limit exists (4), and the limit equals the function value (4 = 4), so all three conditions hold and f is continuous at x = 2.
Common mistakes and how to avoid them
A common error is thinking the limit is always the function value. The limit is what the function approaches, which can differ from f(c), or exist even where f(c) is undefined (a hole). Do not assume they are the same.
Students often stop at 0/0 and conclude the limit does not exist. 0/0 is indeterminate — it means you must do more work (factor, rationalise), not that there is no limit.
Another mistake is forgetting to check both one-sided limits. A two-sided limit exists only if the left and right limits are equal; always check both, especially for piecewise functions.
When testing continuity, remember all three conditions must hold (defined, limit exists, and they are equal). Checking only one is not enough.
Finally, be careful with limits at infinity for rational functions: compare the degrees of numerator and denominator, rather than substituting infinity directly.
Exam technique for "Limits and Continuity"
Always try direct substitution first. If you get a number, that is the limit; if you get 0/0, use factoring, cancelling or rationalising. Show your algebra clearly.
For continuity questions, check all three conditions explicitly and name the type of discontinuity if the function is not continuous. Use one-sided limits for piecewise functions and points where the rule changes.
Connect limits at infinity to horizontal asymptotes and infinite limits to vertical asymptotes. Be ready to apply the Intermediate Value Theorem to justify that a solution exists on an interval. Use correct notation and reasoning, as the AP exam rewards clear justification.
Quick revision summary
- A limit is the value f(x) approaches as x → c, which may differ from f(c) or exist where f(c) is undefined.
- Evaluate limits graphically, numerically, or algebraically (direct substitution first; factor/rationalise for 0/0).
- 0/0 is indeterminate — do more work, don't conclude "no limit".
- A two-sided limit exists only if the left and right one-sided limits are equal.
- Limits at infinity → horizontal asymptotes; infinite limits → vertical asymptotes.
- Continuity at c needs three things: f(c) defined, the limit exists, and they are equal; discontinuities are removable, jump or infinite.
- The Intermediate Value Theorem guarantees intermediate values for a continuous function on [a, b].