How to Solve AP Calculus AB Unit 1 FRQ: Limits & Continuity

The fastest way to solve an AP Calculus AB Unit 1 FRQ is to identify whether the problem is asking about a limit, continuity, or the Intermediate Value Theorem (IVT), then apply the exact procedure the rubric rewards. An AP Calculus AB Unit 1 FRQ is a free-response question from the limits-and-continuity unit: you show work in numbered parts, and College Board awards points for correct setup and justification, not just the final number. Unit 1 covers limits, continuity, and the IVT as the foundation for the rest of the course Wikipedia. This guide breaks down each question type and how to earn every point.

FRQ Format & Scoring

Unit 1 free-response questions are multi-part, usually labeled (a) through (d). Parts build on each other: an early answer often feeds a later justification. The scoring guide assigns points per part, typically one or two points each, and partial credit is generous when the reasoning is visible.

Points are awarded for the procedure and the justification, not only the answer. If you evaluate a limit correctly but skip the algebra, you can lose a point. If you write “discontinuity” without stating which condition fails, you can lose the reasoning point.

Two habits protect your score:

• Show the substitution step or the algebraic simplification, because the rubric looks for it.
• Label each answer with its part letter, so the grader can find your work quickly.

Some Unit 1 questions also include a table of values and ask about a limit. Those questions test whether you can reason from data, not just from a formula.

Solving Limit FRQs

Limit questions in Unit 1 come in three forms: algebraic, graphical, and numerical. The method depends on what the problem gives you.

Algebraic Limits

Direct substitution is always the first attempt. If substituting the input value produces a real number, that number is the limit. If it produces a form like 0/0, the expression needs work before you substitute again.

For 0/0 cases, factor and cancel, rationalize a radical, or combine fractions. For example, to find the limit of (x² − 9)/(x − 3) as x approaches 3, factor the numerator as (x − 3)(x + 3), cancel the (x − 3), and substitute to get 6. Write the cancellation step — that is where the point lives.

Graphical Limits

When the problem supplies a graph, read the limit from the behavior of the curve as x approaches the value from both sides. The limit exists only when the left-hand and right-hand limits agree.

A filled point, an open circle, and a vertical asymptote are three different situations. The value of the function at a point is separate from the limit at that point: the limit concerns nearby behavior, while the function value is what the graph assigns at the input itself. Questions often ask both, precisely to test that distinction.

Numerical Limits

A table of values is approach-only evidence. To estimate a limit numerically, look at the outputs as the inputs get closer from the left and from the right. If both columns converge to the same number, that number is the limit.

One caution: a table can never prove a limit. If a problem asks you to justify a limit from a table, say the values appear to approach a number, and note that both one-sided trends agree.

Continuity & IVT FRQs

A function is continuous at a point when three conditions hold at once: the function value exists, the limit exists, and the two are equal. Discontinuities are classified by which condition breaks — a removable discontinuity (a hole) means the limit exists but does not equal the function value, while a jump or an asymptote means the limit itself fails to exist as a finite number.

For piecewise functions, continuity questions usually hinge on a parameter. To solve them, set the left-hand limit equal to the right-hand limit, and set that common value equal to the function value. That equation gives you the value of the parameter.

The Intermediate Value Theorem

The IVT says that if a function is continuous on a closed interval [a, b], then it takes every value between f(a) and f(b). In FRQs, the standard task is to justify that a solution exists for an equation like f(x) = 0 on a given interval.

A full-credit IVT justification has three parts:

1. State that the function is continuous on the interval.
2. Give two function values with opposite signs, such as f(a) < 0 and f(b) > 0.
3. Conclude that f© = 0 for some c between a and b.

Skipping the continuity statement costs the justification point even when the conclusion is right.

Practice Strategy & Errors

Timed practice beats rereading notes. Work one FRQ at a time under exam timing, then grade yourself against the official scoring guide instead of just checking the answer. Rewrite any lost point in full sentences, because the rubric rewards stated reasoning College Board.

The most common Unit 1 errors:

• Dropping the limit notation partway through the work, which makes a correct answer look unfinished.
• Confusing the function value at a point with the limit at that point.
• Writing a limit that does not exist as a number, instead of stating it does not exist.
• Using the IVT without first stating continuity.
• Cancelling terms before checking whether the denominator is zero.

A short weekly routine works well: two timed FRQs, one self-graded rewrite, and a list of the errors you keep repeating. Review the list before the next set rather than rereading solved problems.

Reading the Question Before You Compute

Most lost points in Unit 1 come from misreading, not miscalculating. Before touching the pencil, underline the verb. “Evaluate” means produce a value with supporting work. “Justify” means write the reason, not just the answer. “Determine whether” means you must commit to a yes-or-no conclusion, so a hedged sentence like “it could be continuous” earns nothing.

Note whether the limit is one-sided or two-sided. A notation like x → 3⁻ asks only about the left side; answering with a two-sided comparison wastes time and can confuse your own work. Similarly, watch for the difference between “the limit as x approaches 2” and “the value at x = 2.” These are separate questions, and a single FRQ often asks both in consecutive parts to see whether you conflate them.

Checking Your Work Under Time

Reserve the last minute of each FRQ to reread your own sentences as if you were the grader. Ask three questions: Did I answer every part? Did I state each justification in words? Did I leave the limit notation in place until the final answer?

A quick numerical sanity check catches many algebra slips. If you claim the limit of a rational expression is 6 as x approaches 3, plug in x = 2.9 and x = 3.1 mentally to confirm the outputs are near 6. This does not replace the algebraic work, but it flags an answer that cannot be right.

Finally, organize the page. Labeled parts with clean steps are faster to grade and easier for you to review.

Key Takeaways

• Identify the question type first: limit, continuity, or IVT.
• Show the substitution or simplification step; the rubric awards process points.
• Algebraic limits: try direct substitution, then factor, rationalize, or combine for 0/0 forms.
• Graphical and numerical limits require the left-hand and right-hand sides to agree.
• Continuity needs three conditions at a point; state which one fails for discontinuities.
• A full IVT answer states continuity, shows opposite-signed values, and concludes a root exists.
• Practice under time pressure and grade yourself against the official scoring guide.