Basic Properties of Limits of Sequences

Given two converging sequences and , where and (with and being real numbers), the following properties hold:
  1. Multiplication by a Constant
    where is a constant
  2. Sum of Sequences
  3. Difference of Sequences
  4. Product of Sequences
  5. Quotient of Sequences
    where and

Calculating Limits of Sequences

  1. Indeterminate Form
    Divide both the numerator and denominator by the highest power of in the denominator.
    1. If the degree of the numerator equals the degree of the denominator, the limit is the ratio of the leading coefficients.
    2. If the degree of the numerator is less than the degree of the denominator, the limit is .
    3. If the degree of the numerator is greater than the degree of the denominator, the sequence diverges to or .
  2. Indeterminate Form
    1. For polynomials, factor out the highest degree term.
    2. For expressions with radicals, rationalize the term with the square root.

Ordering of Limits of Sequences

Given two converging sequences and such that and (with and as real numbers):
  1. If for all natural numbers , then .
  2. If a sequence satisfies for all natural numbers , and if , then:
  3. It is not always true that if , then .
    For example, if and , we have , but .