Topic A: Limits Graphically
Lessons 1–3 · Understanding limits from graphs and tables.
01–03 — Lessons 1–3
Limits from Graphs
A limit describes the value a function approaches as x gets close to a point — not necessarily the value AT the point.
- Notation: lim(x→c) f(x) = L means f(x) approaches L as x → c.
- One-sided limits: lim(x→c⁻) from the left; lim(x→c⁺) from the right.
- A limit exists only if the left-hand and right-hand limits are equal.
- The limit can exist even if f(c) is undefined.
- Limits at infinity: describes end behavior as x → ±∞.
f has a hole at x=2. lim(x→2) f(x) = 5, but f(2) is undefined — the limit still equals 5.
Practice — Lessons 1–3
Q1 · Multiple Choice
lim(x→3) f(x) exists if and only if:
Q2 · Multiple Choice
A function has lim(x→2⁻) f(x) = 4 and lim(x→2⁺) f(x) = 7. Does lim(x→2) f(x) exist?
Topic B: Limits Algebraically
Lessons 4–6 · Direct substitution, factoring, and indeterminate forms.
04–06 — Lessons 4–6
Evaluating Limits Algebraically
- Direct substitution: if f is continuous at c, then lim(x→c) f(x) = f(c).
- Indeterminate form 0/0: factor, cancel, then substitute.
- Rationalize: multiply by conjugate when radicals cause 0/0.
- Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) and both limits equal L, then lim f(x) = L.
lim(x→2) (x²−4)/(x−2) → substitute gives 0/0 → factor: (x+2)(x−2)/(x−2) = x+2 → lim = 4
Practice — Lessons 4–6
Q3 · Short Answer
Find lim(x→3) (x² − 9)/(x − 3) by factoring.
Q4 · Multiple Choice
When direct substitution gives 0/0, this is called:
Topic C: Continuity & Introduction to Derivatives
Lessons 7–9 · Continuity conditions and the concept of a derivative.
07–09 — Lessons 7–9
Continuity & Derivatives
A function is continuous at x = c if: (1) f(c) exists, (2) lim(x→c) f(x) exists, (3) they are equal.
- Types of discontinuity: removable (hole), jump, infinite (asymptote).
- Derivative: the instantaneous rate of change at a point.
- f'(x) = lim(h→0) [f(x+h) − f(x)] / h — the limit of the difference quotient.
- The derivative at a point = the slope of the tangent line to f at that point.
- If f'(x) > 0, f is increasing; if f'(x) < 0, f is decreasing.
f(x) = x². f'(x) = lim(h→0) [(x+h)² − x²]/h = lim(h→0) (2x+h) = 2x.
Practice — Lessons 7–9
Q5 · Multiple Choice
A function is NOT continuous at x = 4 if:
Q6 · Multiple Choice
The derivative f'(c) represents: