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10 multiple-choice questions and 26 flashcards on Limits and Continuity, about 7% of the High School Calculus bank. Every one carries a written rationale.
Limits and Continuity is one of 8 chapters in CoStudy's High School Calculus bank, and it holds 10 of the bank's 150 multiple-choice questions — roughly 7% of the total. That proportion is not arbitrary: chapters follow the certifying body's published exam outline, and the number of questions in each is set by that domain's published weight, so the share of your practice time this chapter takes matches the share of the real exam it accounts for.
Studying by chapter is worth doing once you have a diagnostic score. A single overall percentage tells you whether you are close; it does not tell you which domain is dragging. Working a weak chapter in isolation, and re-testing it in isolation, is the fastest way to move a score that has stalled — and it is why the mock exams in CoStudy report by domain rather than as one number.
4 questions drawn from this chapter, with the full rationale shown — the controlling principle behind the right answer, and why each wrong option tempts and fails.
lim(x→3) (x² − 9)/(x − 3) =
Answer: C — 6
C) Factor: (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3 → limit = 6. A) Used direct substitution and stopped at 0/0. B) Computed numerator value 9 at x = 3. D) Used x = 3 in factored form without adding.
Evaluate ∫₀² 2x dx.
Answer: C — 4 (antiderivative x²; evaluate at bounds: 2² - 0² = 4)
Definite integral: ∫₀² 2x dx = [x²]₀² = 4 - 0 = 4. Fundamental Theorem of Calculus. Represents area under curve y = 2x from 0 to 2 (right triangle area).
If f(x) = 3x² + 4x + 1, find f'(x).
Answer: A — 6x + 4 (power rule on each term)
Differentiate term-by-term using power rule. d/dx[3x²] = 6x. d/dx[4x] = 4. d/dx[1] = 0. Sum: 6x + 4. Linearity of derivative.
Find d/dx[sin(x)].
Answer: E — cos(x) (derivative of sine is cosine)
Common derivatives: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x), d/dx[eˣ] = eˣ, d/dx[ln(x)] = 1/x. Memorize.
4 cards from the 26 in this chapter.
Evaluate lim_{x→0} x² sin(1/x).
By squeeze: |x² sin(1/x)| ≤ x² → 0. So limit = 0.
Use L'Hôpital: lim_{x→0} sin x / x.
lim cos x / 1 = 1.
Evaluate lim_{x→∞} 1/x.
0.
If f is differentiable at a, is f continuous at a?
Yes. Differentiability implies continuity (the converse is not always true).
These are a sample. The full Limits and Continuity chapter runs 36 items with per-chapter progress tracking, on the web and in the iOS app.
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