Calculus (Subtest III)
Limits and Continuity
Calculus on Subtest III spans limits, differentiation, integration, and series. A limit exists when the left- and right-hand limits agree; continuity at a requires f(a) defined, the limit existing, and the two being equal. Standard limits include lim_{x→0}(sin x)/x = 1 and lim_{x→0}(1 − cos x)/x² = 1/2. L'Hôpital's rule applies only to indeterminate forms 0/0 or ∞/∞, and only after verifying the form.
Differentiation
The definition f′(x) = lim_{h→0}[f(x+h) − f(x)]/h underlies the power, product, quotient, and chain rules. Use implicit differentiation when y is not isolated, and logarithmic differentiation for variable exponents. Derivatives drive curve analysis: f′ > 0 increasing, f″ > 0 concave up, critical points where f′ = 0 or fails, and inflection where concavity changes.
Integration and Series
The Fundamental Theorem links area and antiderivatives: ∫_a^b f(x)dx = F(b) − F(a). Techniques include u-substitution, integration by parts, and partial fractions; applications include area between curves, volumes of revolution, and average value. For series, know geometric convergence (|r| < 1), the p-series test, ratio and root tests, and alternating series; power series converge on an interval whose endpoints must be checked separately.
Worked Example
Evaluate lim_{x→0}(1 − cos x)/x². Direct substitution gives 0/0, so apply L'Hôpital once: the derivative of the numerator is sin x and of the denominator 2x, giving lim (sin x)/(2x) = (1/2)·lim (sin x)/x = 1/2. Alternatively, multiply numerator and denominator by (1 + cos x) to get (sin²x)/(x²(1 + cos x)) = [(sin x)/x]² · 1/(1 + cos x) → 1 · 1/2 = 1/2. Both routes agree.
Traps That Cost Marks
- Applying L'Hôpital to a form that is not indeterminate.
- Omitting +C on indefinite integrals or forgetting to change limits after substitution.
- Dropping the chain rule in composite functions.
- Confusing convergence of a sequence with convergence of its series.
- Checking endpoints of a power series interval with the wrong test or not at all.