Let (an)n=1∞ and (bn)n=1∞ be convergent sequences of real numbers, and let x,y be the real numbers given by x=n→∞liman and y=n→∞limbn. Then we have the following laws. In the quotient law, assume y=0 and that bn=0 for all sufficiently large n.
n→∞lim(an±bn)=n→∞liman±n→∞limbn=x±y,n→∞lim(anbn)=n→∞liman⋅n→∞limbn=xy,n→∞limbnan=n→∞limbnn→∞liman=yx(if bn and y are nonzero),n→∞limmax{an,bn}=max{n→∞liman,n→∞limbn}=max{x,y},n→∞limmin{an,bn}=min{n→∞liman,n→∞limbn}=min{x,y}.
Limit laws for sequences describe how convergence behaves under algebraic operations: the limit of a sum, product or quotient of convergent sequences is the sum, product or quotient of their limits. They allow limits of complicated sequences to be inferred from simpler ones without returning to the ε-N definition each time.
A typical use is a quotient of polynomials in n, such as
an=n2+2n3n2+1.
Numerator and denominator both grow without bound, so the quotient law cannot be applied directly. After dividing both by n2,an=(3+1/n2)/(1+2/n), and now every part converges: 3+1/n2→3 and 1+2/n→1=0, so an→3.
The hypotheses matter. Every law needs both sequences to converge. If an and bn both grow without bound, an−bn can do anything: (n+1)−n=1 for every n, while n2−n grows without bound. The quotient law also needs y=0: with an=1 and bn=1/n the quotient an/bn=n diverges, while with an=bn=1/n it equals 1.
The laws only work in one direction. Convergence of an+bn does not imply convergence of an and bn:(−1)n+(−1)n+1=0 for every n. When one factor diverges, the product law gives no information, yet the limit may exist: (−1)n⋅n1→0 by the squeeze theorem. The laws for max and min follow from the others through max{a,b}=21(a+b+∣a−b∣) and min{a,b}=21(a+b−∣a−b∣), since ∣an−bn∣→∣x−y∣ by the reverse triangle inequality.