The intermediate value theorem (IVT) says that if f is continuous on a closed interval [a, b] and N is any number between f(a) and f(b), then f(c) = N for at least one c between a and b. Its most common use: if f(a) and f(b) have opposite signs, f has a zero between a and b.
A function is continuous at x = a when its graph has no break there. In calculus, continuity at a requires three conditions: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a). If any condition fails, f has a discontinuity at a: removable, jump or infinite.
The squeeze theorem finds a limit by trapping a function between two others. If g(x) ≤ f(x) ≤ h(x) for x near a, and g(x) and h(x) both approach L, then f(x) approaches L too. It is also called the sandwich theorem.
A limit at infinity describes what f(x) approaches as x grows without bound, written as the limit of f(x) as x → ∞ or x → −∞. If the limit is a number L, the line y = L is a horizontal asymptote. For rational functions, divide every term by the highest power of x in the denominator.
An infinite limit describes a function whose values grow without bound as x approaches a number. We write lim f(x) = ∞ (or −∞) when f(x) becomes larger (or more negative) than any number you choose. If either one-sided limit at a is infinite, the line x = a is a vertical asymptote.
Evaluating limits algebraically means finding a limit exactly instead of estimating it. First try direct substitution. If that gives a number, it is the limit. If it gives 0/0, rewrite the expression by factoring and canceling, multiplying by a conjugate, or combining fractions, then substitute again.
The limit laws let you break a hard limit into easy pieces. If the limits of f(x) and g(x) as x approaches a both exist, the limit of a sum, difference, constant multiple, product, quotient (when the bottom limit is not 0), power or root equals that same combination of the separate limits.