Stewart Calculus · 8th Edition
Chapter 2: Limits & Derivatives
Every key formula from Stewart Calculus Chapter 2, in one searchable page. Click a card to study it — worked examples included.
2.1–2.2 · TANGENTS & LIMITS
Tangent slope
m = limh→0 f(a+h) − f(a)h
Instantaneous velocity v(a) uses the same quotient with position s(t).
One-sided limits
limx→a f(x) = L ⟺ left and right limits both equal L
KEY NOTES
- A limit asks where f(x) heads — the value at x = a itself never matters.
2.3 · LIMIT LAWS
If limx→a f(x) = L and limx→a g(x) = M, then
lim(f ± g) = L ± M · lim(fg) = LM · lim(f/g) = L/M (M ≠ 0)
lim[f(x)]n = Ln · lim n√f(x) = n√L
lim[f(x)]n = Ln · lim n√f(x) = n√L
Direct substitution
Polynomials — and rationals with Q(a) ≠ 0: just plug in x = a.
Squeeze Theorem
f ≤ g ≤ h near a, outer limits L ⇒ limx→a g(x) = L
KEY NOTES
- Stuck on 0/0? Do algebra first: factor & cancel, or rationalize.
WORKED EXAMPLE
limx→3 x² − 9x − 3 is 0/0 — factor: (x−3)(x+3)x − 3 = x + 3 (x ≠ 3).
⇒ 3 + 3 = 6.
⇒ 3 + 3 = 6.
2.4 · THE PRECISE DEFINITION
limx→a f(x) = L means: for every ε > 0 there is δ > 0 such that
0 < |x − a| < δ ⇒ |f(x) − L| < ε
0 < |x − a| < δ ⇒ |f(x) − L| < ε
KEY NOTES
- δ depends on ε — a tighter target may need a tighter x-window.
2.5 · CONTINUITY
f is continuous at a — all three hold
(1) f(a) is defined · (2) limx→a f(x) exists · (3) limx→a f(x) = f(a)
Intermediate Value Theorem
f continuous on [a, b], N between f(a) and f(b)
⇒ some c ∈ (a, b) has f(c) = N.
⇒ some c ∈ (a, b) has f(c) = N.
WORKED EXAMPLE
x³ − x − 1 = 0 has a root in (1, 2): f is continuous,
f(1) = −1 < 0 < 5 = f(2) ⇒ IVT gives c ∈ (1,2) with f(c) = 0.
f(1) = −1 < 0 < 5 = f(2) ⇒ IVT gives c ∈ (1,2) with f(c) = 0.
limx→∞ 1xr = 0 (r > 0)
Asymptotes
y = L horizontal if limx→±∞ f(x) = L · x = a vertical if limx→a f(x) = ±∞
WORKED EXAMPLE
limx→∞ 3x² + 12x² − 5: divide top & bottom by x² →
3 + 1/x²2 − 5/x² → 32.
3 + 1/x²2 − 5/x² → 32.
KEY NOTES
- Rational end behavior: deg top < deg bottom → 0; equal → ratio of leading coefficients; top > bottom → ±∞.
2.7–2.8 · THE DERIVATIVE
f′(a) = limh→0 f(a+h) − f(a)h = limx→a f(x) − f(a)x − a
f′(x) = limh→0 f(x+h) − f(x)h
f′(a) = slope of the tangent at a = instantaneous rate of change.
KEY NOTES
- Differentiable ⇒ continuous — but not conversely: corners, cusps & vertical tangents break it.
WORKED EXAMPLE
f(x) = x²: f′(3) = limh→0 (3+h)² − 9h
= limh→0 6h + h²h = limh→0 (6 + h) = 6.
= limh→0 6h + h²h = limh→0 (6 + h) = 6.
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