AP Calculus AB and BC glossary
Limit laws
The limit laws let you distribute a limit across sums, differences, products, quotients, and powers. Each law requires the individual limits to exist, and the quotient law additionally requires the denominator's limit to be nonzero.
These are what justify direct substitution for continuous functions. Because polynomials are built from sums and products of continuous pieces, their limits can always be evaluated by plugging in.
The conditions are not decoration
Splitting a limit when one piece does not exist can produce a wrong answer. Applied to an indeterminate form, the quotient law simply does not apply.
Appears in: Unit 1: Limits and Continuity