algebra of limits
The algebra of limits is the set of rules that lets you compute complicated limits by breaking them into simple pieces. If you already know the limits of two sequences, you can read off the limits of their sum, difference, product, and quotient without going back to epsilon-N — the limit operation politely passes through ordinary arithmetic.
Precisely, suppose a_n -> A and b_n -> B. Then a_n + b_n -> A + B, a_n - b_n -> A - B, and a_n b_n -> AB; for any constant c, c a_n -> cA. For quotients, a_n / b_n -> A / B provided B is not 0 (and the terms b_n are eventually nonzero). These rules are theorems proved from the epsilon-N definition, typically via the triangle inequality, not mere notation.
Two cautions keep these rules honest. First, every rule presumes both limits already exist (and are finite); you may not split a limit across operations until you know the pieces converge. Second, the quotient rule genuinely needs B nonzero — when B = 0 the ratio can do anything, which is the origin of indeterminate forms like 0/0 that require separate tools such as the squeeze theorem or careful estimation.
To evaluate (3n^2 + 2)/(n^2 - n), divide top and bottom by n^2: (3 + 2/n^2)/(1 - 1/n). Now 2/n^2 -> 0 and 1/n -> 0, so by the algebra of limits the ratio tends to (3 + 0)/(1 - 0) = 3.
Reduce to known limits like 1/n -> 0, then combine with the limit laws.
Splitting a limit when one piece diverges is a classic error: a_n = n and b_n = -n both diverge, yet a_n + b_n = 0 converges. The sum rule simply does not apply when the parts have no finite limits.