Limits, continuity and differentiation of real functions of one variable
Limits, continuity and differentiation of real functions of one variable
This is the largest block of the paper: a limit or continuity question with a parameter, and a full analysis of a function (domain, asymptotes, monotonicity, extrema, curvature) or a Rolle/mean value theorem proof.
Core rules
- Indeterminate forms: L'Hopital applies only to 0/0 and infinity/infinity, differentiating numerator and denominator separately. Convert 0.infinity, infinity - infinity and 1^infinity (by logarithms or by writing f^g = e^(g ln f)) before applying it.
- Equivalent infinitesimals as x tends to 0: sin x ~ x, tan x ~ x, 1 - cos x ~ x^2/2, ln(1 + x) ~ x, e^x - 1 ~ x, (1 + x)^a - 1 ~ ax.
- Continuity at a: f(a) is defined, the limit exists and both agree. In piecewise functions, force the two one-sided limits and the value to coincide.
- Differentiability implies continuity but not the converse; at a point the two one-sided derivatives must agree.
- Rolle: if f is continuous on [a,b], differentiable on (a,b) and f(a) = f(b), then f'(c) = 0 for some c. Mean value theorem: f'(c) = (f(b) - f(a))/(b - a). Combined with Bolzano, they prove that an equation has exactly one root: existence by a sign change, uniqueness by monotonicity.
- Shape: f' > 0 means increasing; f'' > 0 means convex. Extrema occur where f' = 0 or f' does not exist, confirmed by a sign change of f'. Vertical asymptotes where the limit is infinite, horizontal and oblique ones through limits of f(x)/x.
Worked example
Compute the limit of (e^x - 1 - x)/x^2 as x tends to 0. Substitution gives 0/0, so L'Hopital applies: (e^x - 1)/(2x), still 0/0, so apply it again to get e^x/2, which tends to 1/2. The Maclaurin expansion e^x = 1 + x + x^2/2 + ... gives the same answer and checks the work.
Traps that cost marks
Using L'Hopital on a limit that is not indeterminate; differentiating a quotient with L'Hopital when the quotient rule is needed; assuming that continuity implies differentiability; forgetting to check f(a) = f(b) before invoking Rolle; ignoring the domain of ln x or sqrt(x); and forgetting that extrema can also occur at endpoints of the interval.