Limits of a function — A/L Combined Mathematics

What must I remember about Limits of a function?

• Intuitive and formal concept of limit: lim_x->a f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to a (x ≠ a); two-sided limit exists if and only if left-hand limit lim_x->a^- f(x) equals right-hand limit lim_x->a⁺ f(x) = L.
• Limit laws: lim (f ± g) = lim f ± lim g; lim (f * g) = (lim f) * (lim g); lim (f / g) = (lim f) / (lim g) provided lim g ≠ 0; lim [k * f(x)] = k * lim f(x).
• Indeterminate forms: 0/0, ∞/∞, 0 * ∞, ∞ - ∞; resolved by algebraic factoring, rationalizing surds with conjugate expressions, or dividing by highest power of x as x -> ∞.
• Standard algebraic and trigonometric limits: lim_x->a (xⁿ - aⁿ) / (x - a) = n * a^n-1 (for rational n); lim_x->0 (sin x) / x = 1 (x in radians); lim_x->0 (tan x) / x = 1; lim_x->0 (1 - cos x) / x² = 1/2.
• Continuity definition: A function f(x) is continuous at x = a if and only if: 1) f(a) is defined, 2) lim_x->a f(x) exists, and 3) lim_x->a f(x) = f(a).

Test yourself (1 minute)

1. Sandwich / Squeeze Theorem

   If g(x) ≤ f(x) ≤ h(x) in an open interval around a, and lim_x->a g(x) = lim_x->a h(x) = L, then lim_x->a f(x) = L; foundational proof for lim_x->0 (sin x)/x = 1 using unit circle geometric areas.

2. Limits at infinity for rational functions

   For P(x)/Q(x) as x -> ∞: if deg(P) < deg(Q), limit is 0; if deg(P) = deg(Q), limit is ratio of leading coefficients; if deg(P) > deg(Q), limit diverges to ±∞.

3. Radian requirement pitfall

   Formula lim_x->0 (sin x)/x = 1 holds ONLY when x is measured in radians; if x is in degrees, the limit is π/180.

Practise Limits of a function questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 13

What should I study first?

What comes next?

Next in the syllabus

Where does Limits of a function fit in the syllabus?

Year 1 · Term 2