Functions: Limits, Continuity and Differentiation (Optimisation and Tangent Lines)
This domain tests limits, asymptotes, continuity (including piecewise functions with a parameter), derivative rules, the tangent line, monotonicity and extrema, and optimisation of functions taken from economic or social contexts.
Limits and asymptotes
- A 0/0 indeterminate form is resolved by factoring and cancelling, or by L'Hôpital's rule, which is valid only for 0/0 and ∞/∞.
- For a rational function as x → ±∞, compare degrees: equal degrees give the ratio of leading coefficients; numerator of lower degree gives 0; numerator of higher degree gives ±∞.
- A vertical asymptote appears at values where the denominator vanishes and no factor cancels.
Continuity
f is continuous at x = a when the two one-sided limits exist, are equal, and equal f(a). For a piecewise function, the parameter is found by imposing lim(x→a⁻) f = lim(x→a⁺) f = f(a) at the joining point.
Derivative and its applications
- (xⁿ)' = n·xⁿ⁻¹, plus the product, quotient and chain rules.
- Tangent at x = a: y − f(a) = f'(a)(x − a). Both f(a) and f'(a) are needed.
- f' > 0 → increasing, f' < 0 → decreasing. A critical point f'(a) = 0 is an extremum only if f' changes sign there; alternatively f''(a) > 0 gives a minimum and f''(a) < 0 a maximum.
Worked example
f(x) = x³ − 6x² + 9x + 1. Then f'(x) = 3x² − 12x + 9 = 3(x − 1)(x − 3). The sign of f' is + on (−∞,1), − on (1,3) and + on (3,∞), so f increases, decreases and increases: local maximum at x = 1 with f(1) = 5, local minimum at x = 3 with f(3) = 1. The tangent at x = 0 is y = f(0) + f'(0)x = 1 + 9x, since f(0) = 1 and f'(0) = 9.
Traps that cost marks
- Using L'Hôpital on a limit that is not indeterminate, or on a form such as 0·∞ without rewriting it first.
- Assuming every point with f'(a) = 0 is an extremum (x³ at 0 is the standard counterexample).
- Giving the tangent as y = f'(a)(x − a) and forgetting the +f(a) term.
- In optimisation word problems, not restricting the domain to the values the context allows (x ≥ 0, capacity limits) and not stating which value is the maximum and what it means.