A Limit Process

While computing $\displaystyle\lim_{x\to\alpha}R(x)$ for certain functions $R(x)$, direct substitution can produce one of the following forms:

These are said to have "the form of a number," but no value can be assigned to them consistently with the ordinary rules of addition and multiplication — hence the name indeterminate forms. Although not themselves numbers, they play a central role in the limiting behaviour of a function, since resolving which actual number (if any) the limit approaches requires more than substitution. …