Indeterminate Form - Discussion

Discussion

The most common example of an indeterminate form is 0/0. As x approaches 0, the ratios x/x3, x/x, and x2/x go to, 1, and 0 respectively. In each case, however, if the limits of the numerator and denominator are evaluated and plugged into the division operation, the resulting expression is 0/0. So (roughly speaking) 0/0 can be 0, or, or it can be 1 and, in fact, it is possible to construct similar examples converging to any particular value. That is why the expression 0/0 is indeterminate.

More formally, the fact that the functions f and g both approach 0 as x approaches some limit point c is not enough information to evaluate the limit

That limit could converge to any number, or diverge to infinity, or might not exist, depending on what the functions f and g are.

In some theories a value may be defined even where the function is discontinuous. For example |x|/x is undefined for x = 0 in real analysis. However it is the sign function with sgn(0) = 0 when considering Fourier series or hyperfunctions.

Not every undefined algebraic expression is an indeterminate form. For example, the expression 1/0 is undefined as a real number but is not indeterminate. This is because any limit that gives rise to this form will diverge to infinity.

An expression representing an indeterminate form may sometimes be given a numerical value in settings other than the computation of limits. The expression 00 is defined as 1 when it represents an empty product. In the theory of power series, it is also often treated as 1 by convention, to make certain formulas more concise. (See the section "Zero to the zero power" in the article on exponentiation.) In the context of measure theory, it is usual to take to be 0.