Limits, continuity and differentiation of real functions of one variable
Limits, continuity and differentiation of real functions of one variable
This shared analysis guide practises limits, continuity, derivatives, function behavior and theorem conditions. The practice scope is not a verified statement about the current regional weighting or choice structure.
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 require an infinite one-sided limit. Horizontal asymptotes use lim f(x); an oblique asymptote y = mx + b uses finite m = lim f(x)/x and b = lim(f(x) - mx), with m nonzero.
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.