Limits & Continuity Cheat Sheet

⚖️ Fundamental Limit Laws

Assume lim f(x) = L and lim g(x) = M exist as x→a:

lim [f(x) ± g(x)] = L ± M

lim [f(x) · g(x)] = L · M

lim [f(x) / g(x)] = L / M (M ≠ 0)

lim [c · f(x)] = c · L

lim [f(x)]ⁿ = Lⁿ

Example: lim (x² + 3x) = lim x² + lim 3x = 4 + 6 = 10 (x→2)

🔄 Continuity Conditions

A function f(x) is continuous at x = a if:

Discontinuity Types:
1. Removable (hole)
2. Jump
3. Infinite

If f and g are continuous at a, then f±g, f·g, f/g (g(a)≠0) are continuous

✨ Essential Trigonometric Limits

limₓ→₀ [sin(x)/x] = 1

limₓ→₀ [tan(x)/x] = 1

limₓ→₀ [(1 - cos(x))/x] = 0

limₓ→₀ [(1 - cos(x))/x²] = 1/2

Application: lim (sin 3x)/x = 3·lim (sin 3x)/(3x) = 3·1 = 3

Crucial for deriving trig derivatives

🏥 L'Hôpital's Rule

For indeterminate forms 0/0 or ∞/∞:

lim [f(x)/g(x)] = lim [f'(x)/g'(x)]

Requirements:
1. f(a) = g(a) = 0 or ±∞
2. f and g differentiable near a
3. g'(x) ≠ 0 near a (except possibly at a)

Example: lim (eˣ - 1)/x = lim eˣ/1 = 1 (x→0)

🥪 Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near a and lim g(x) = lim h(x) = L:

Then lim f(x) = L

Classic: Show lim x² sin(1/x) = 0 using -x² ≤ x² sin(1/x) ≤ x²

Useful when direct substitution fails

∞ Limits at Infinity

limₓ→∞ [1/x] = 0

limₓ→∞ [aₙxⁿ + ...]/[bₘxᵐ + ...] = {0 if n < m
aₙ/bₙ if n = m
±∞ if n > m}

limₓ→∞ eˣ = ∞

limₓ→-∞ eˣ = 0

ε-δ Definition

lim f(x) = L as x→a means:

∀ε > 0, ∃δ > 0 such that
0 < |x - a| < δ ⇒ |f(x) - L| < ε

Strategy:
1. Start with |f(x) - L| < ε
2. Work backwards to find δ
3. Verify your δ works

↔ One-Sided Limits

limₓ→a⁺ f(x) = L⁺ (right-hand limit)

limₓ→a⁻ f(x) = L⁻ (left-hand limit)

limₓ→a f(x) exists iff L⁺ = L⁻

Example:
f(x) = |x|/x
limₓ→0⁺ = 1, limₓ→0⁻ = -1
∴ limit does not exist