Module 1 Quick Reference

Limits and Continuity – Calculus I

Limit Definition & One-Sided Limits

limx→a f(x) = L  exists ⇔  limx→a− f(x) = limx→a+ f(x) = L

Limit DNE if: one-sided limits differ, function blows up, or function oscillates.

Limit Laws

lim [f ± g] = lim f ± lim g

lim [f · g] = (lim f)(lim g)

lim [cf] = c · lim f

lim [f/g] = (lim f)/(lim g), lim g ≠ 0

lim [f]n = (lim f)n

lim √f = √(lim f)

Computing Limits: Strategy

1. Direct substitution — if no 0/0, done.

2. 0/0 form: Factor, rationalize, or rewrite.

3. nonzero/0: Limit is ±∞ (check signs).

4. Trig: Use (sin x)/x → 1.

5. Squeeze Theorem for bounded oscillation.

Special Limits

limx→0 (sin x)/x = 1     limx→0 (1 − cos x)/x = 0

Limits at Infinity (Rational Functions)

deg(num) < deg(den)

Limit = 0

deg(num) = deg(den)

Limit = leading coeff ratio

deg(num) > deg(den)

Limit = ±∞ (no horizontal asymptote)

Asymptotes

Horizontal: y = L

limx→±∞ f(x) = L

Vertical: x = a

Denominator → 0, numerator ≠ 0

Continuity

f continuous at a ⇔ (1) f(a) defined, (2) lim exists, (3) lim = f(a)

Removable

Limit exists, f(a) wrong/missing

Jump

One-sided limits differ

Infinite

At least one side → ±∞

Intermediate Value Theorem

f continuous on [a,b], N between f(a) and f(b) ⇒ ∃ c ∈ (a,b) with f(c) = N

Use for root-finding: show f changes sign on [a,b] ⇒ root exists.