Limits, continuity and differentiation
Limits, continuity and differentiation form the calculus core of the paper: compute limits and remove indeterminacies, decide continuity (often with a parameter in a piecewise function), differentiate, and write the tangent line or interpret a rate of change.
Core rules
- Indeterminacies 0/0, ∞/∞, ∞ − ∞, 0·∞ and 1^∞ are resolved by factorising, using a common denominator, multiplying by the conjugate, dividing by the highest power, or L'Hôpital when the form really is 0/0 or ∞/∞.
- Vertical asymptote at x = a when the denominator vanishes and the factor does not cancel; horizontal asymptote y = L when the limit at ±∞ is L; oblique asymptote y = mx + n with m = lim f(x)/x and n = lim (f(x) − mx).
- f is continuous at a when f(a) exists, lim f(x) exists and both coincide. Discontinuities are avoidable, jump (different one-sided limits) or essential.
- (uv)' = u'v + uv', (u/v)' = (u'v − uv')/v², and the chain rule (f(g(x)))' = f'(g(x))g'(x).
- Tangent line at x = a: y = f(a) + f'(a)(x − a). The derivative is the instantaneous rate of change, so marginal cost is C'(q).
Worked example
For f(x) = x e^{−x}: f(0) = 0 and f'(x) = e^{−x} − x e^{−x} = e^{−x}(1 − x), so f'(0) = 1 and the tangent at x = 0 is y = x. At infinity the exponential dominates, so lim x→+∞ x e^{−x} = 0 and y = 0 is a horizontal asymptote on the right; as x→−∞ the function tends to −∞, so there is no asymptote on that side. The derivative also shows f increases for x < 1 and decreases for x > 1, the extra information a sketch question will want.
Traps that cost marks
- Using L'Hôpital on a form that is not indeterminate, such as 2/0.
- Forgetting that continuity needs all three conditions, or checking only one side of a piecewise join.
- Cancelling a factor and still claiming a vertical asymptote there.
- Giving the normal line when the tangent was asked, or omitting f(a).
- Differentiating a product term by term without the product rule.