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AP Calculus Limits Practice Problems

Unit 1 of AP Calc AB, with answers and worked solutions.

Limits are the first unit of AP Calculus AB and the foundation for everything that follows — the derivative and the definite integral are both defined as limits. On the exam, limit questions are mostly about choosing the right technique, and that choice follows a short checklist worth memorizing:

Substitute first. If plugging in the value gives a real number, that number is the limit — you're done.
Got 0/0? That's an indeterminate form, not an answer. Factor and cancel, or multiply by a conjugate to rationalize, then substitute again.
Absolute values and piecewise functions change formula at a point — check the left and right limits separately. If they disagree, the limit does not exist.
Nonzero over zero means a vertical asymptote: decide the sign of the denominator on each side to get +∞ or −∞.
Near 0, lean on the special trig limit sin(x)/x → 1.

The eight problems below walk through that checklist in order, ending with continuity — a function is continuous at a point exactly when the limit exists, the function value exists, and the two agree. Answer each one and you'll get the worked solution, the same way Lodera's full question bank does it.

Practice problems

Problem 1

x \to 3 lim x + 12

Problem 2

x \to 7 lim \frac{x ^{2} - 49}{x - 7}

Problem 3

x \to 16 lim \frac{x - 16}{x - 4}

Problem 4

x \to 0 lim \frac{sin ( 6 x )}{2 x}

Problem 5

x \to - 5 lim \frac{∣ x + 5 ∣}{x + 5}

Problem 6

x \to 3^{+} lim \frac{x - 5}{x - 3}

Problem 7

f (x) = {3 x - 6 k x^{2} + 4 x < 2 x \geq 2

What value of

k

makes

f

continuous at

x = 2

Problem 8

lim_{x \to 3} f (x) = 2

and

lim_{x \to 3} g (x) = 0

, what can you conclude about

x \to 3 lim \frac{f ( x )}{g ( x )}

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