Indeterminate form

In calculus and other branches of mathematical analysis, when the limit of the sum, difference, product, quotient or power of two functions is taken, it may often be possible to simply add, subtract, multiply, divide or exponentiate the corresponding limits of these two functions respectively. However, there are occasions where it is unclear what the sum, difference, product, quotient, or power of these two limits ought to be. For example, it is unclear what the following expressions ought to evaluate to:

${\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,~0^{0},~1^{\infty },{\text{ and }}\infty ^{0}.}$

These seven expressions are known as indeterminate forms. More specifically, such expressions are obtained by naively applying the algebraic limit theorem to evaluate the limit of the corresponding arithmetic operation of two functions, yet there are examples of pairs of functions that after being operated on converge to 0, converge to another finite value, diverge to infinity or just diverge. This inability to decide what the limit ought to be explains why these forms are regarded as indeterminate. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity). The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.

The most common example of an indeterminate form is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by ${\displaystyle 0/0}$. For example, as ${\displaystyle x}$ approaches ${\displaystyle 0~}$, the ratios ${\displaystyle x/x^{3}}$, ${\displaystyle x/x}$, and ${\displaystyle x^{2}/x}$ go to ${\displaystyle \infty }$, ${\displaystyle 1}$, and ${\displaystyle 0~}$ respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is ${\displaystyle 0/0}$, which is indeterminate. In this sense, ${\displaystyle 0/0}$ can take on the values ${\displaystyle 0~}$, ${\displaystyle 1}$, or ${\displaystyle \infty }$, by appropriate choices of functions to put in the numerator and denominator. A pair of functions for which the limit is any particular given value may in fact be found. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, ${\displaystyle x\sin(1/x)/x}$.

So the fact that two functions ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ converge to ${\displaystyle 0~}$ as ${\displaystyle x}$ approaches some limit point ${\displaystyle c}$ is insufficient to determinate the limit

${\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}.}$

An expression that arises by ways other than applying the algebraic limit theorem may have the same form of an indeterminate form. However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits. An example is the expression ${\displaystyle 0^{0}}$. Whether this expression is left undefined, or is defined to equal ${\displaystyle 1}$, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that ${\displaystyle 0^{\infty }}$ and other expressions involving infinity are not indeterminate forms.