Limits & Continuity

the foundation of everything in calculus 🎯

💡 what is a limit?

lim_x→c f(x) = L means: you can make f(x) as close to L as you want by taking x close enough to c (not necessarily equal to c).

One-sided limits: x → c⁺ (from the right) and x → c⁻ (from the left). A two-sided limit exists only if both one-sided limits exist and agree.

📐 limit laws (for limits that exist)

Sum/difference: limit of (f ± g) = limit f ± limit g
Product: limit of (f·g) = (limit f)(limit g)
Quotient: limit of f/g = (limit f)/(limit g) if limit g ≠ 0
Composition: if g is continuous at L and lim f = L, then lim g(f(x)) = g(L)

lim_x→c k = k · lim_x→c x = c

✅ continuity at x = c

f is continuous at c when: f(c) is defined, lim_x→c f(x) exists, and lim_x→c f(x) = f(c).

Types of discontinuity: removable (hole), jump (left ≠ right), infinite (vertical asymptote behavior).

Intermediate Value Theorem: if f is continuous on [a,b] and k is between f(a) and f(b), then ∃ c in (a,b) with f(c) = k.

🍋 squeeze theorem

If g(x) ≤ f(x) ≤ h(x) near c (except maybe at c) and lim g = lim h = L, then lim f = L. Classic use: limits involving sin x / x as x → 0.

🔥 indeterminate forms & L'Hôpital BC

0/0 or ∞/∞ often mean you can try L'Hôpital: if lim f/g is indeterminate and f,g differentiable near c, then lim f/g = lim f′/g′ (when the latter limit exists).

Other indeterminates (0·∞, ∞−∞, 0⁰, 1^∞, ∞⁰) → rewrite to a fraction or use ln, then apply L'Hôpital if you get 0/0 or ∞/∞.

✏️ quick practice

Find lim_x→2 (x² − 4)/(x − 2).
Explain why f(x) = 1/x has no limit as x → 0.
Use the IVT to show x³ − x − 1 = 0 has a root between 1 and 2.

home unit 2 →

you've got this ⭐