Indeterminate form

In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form. The indeterminate forms include 00, 0/0, 1, ∞ − ∞, ∞/∞, 0 × ∞, and ∞0.

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 $\scriptstyle\infty$, 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 $\scriptstyle\infty$, 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

$\lim_{x \to c} \frac{f(x)}{g(x)}. \!$

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[clarification needed] 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 power of zero" in the article on exponentiation.) In the context of measure theory, it is usual to take $\scriptstyle 0\cdot\infty$ to be 0.

Some examples and non-examples

The form 0/0

The indeterminate form 0/0 is particularly common in calculus because it often arises in the evaluation of derivatives using their limit definition.

As mentioned above,

$\lim_{x \to 0} \frac{x}{x} = 1, \! ~~ (1)$

while

$\lim_{x \to 0} \frac{x^{2}}{x} = 0, \! ~~ (2)$

This is enough to show that 0/0 is an indeterminate form. Other examples with this indeterminate form include

$\lim_{x \to 0} \frac{\sin(x)}{x} = 1, \! ~~ (3)$

and

$\lim_{x \to 49} \frac{x - 49}{\sqrt{x}\, - 7} = 14, ~~ (4)$

Direct substitution of the number that x approaches into any of these expressions leads to the indeterminate form 0/0, but the limits take many different values. In fact, any desired value A can be obtained for this indeterminate form as follows:

$\lim_{x \to 0} \frac{Ax}{x} = A . \! ~~ (5)$

Furthermore, the value infinity can also be obtained (in the sense of divergence to infinity):

$\lim_{x \to 0} \frac{x}{x^3} = \infty . \! ~~ (6)$

The form 00

The indeterminate form 00 has been discussed since at least 1834.[1] The following examples illustrate that the form is indeterminate:

$\lim_{x \to0^+} x^0 = 1 , \! ~~ (7)$

$\lim_{x \to0^+} 0^x = 0. \! ~~ (8)$

Thus, in general, knowing that $\scriptstyle\lim_{x \to c} f(x) \;=\; 0^+\!$ and $\scriptstyle\lim_{x \to c} g(x) \;=\; 0$ is not sufficient to calculate the limit

$\lim_{x \to c} f(x)^{g(x)}.\!$

If the functions f and g are analytic and f is not identically zero in a neighbourhood of c on the complex plane, then the limit of f(z) g(z) will always be 1. This also holds for real functions, but f must not be negative in the domain of the limit; alternatively, f can be the absolute value of an analytic function.[2]

If the pair (f(x), g(x)) remains between two lines y = mx and y = Mx, where m and M are positive numbers, as x approaches c and f(x) and g(x) approaches 0, then the limit is always 1.

In many settings other than when evaluating limits 00 is taken to be defined as 1 even though it is an indeterminate form; see the section zero to the power of zero in the article on exponentiation. One justification for this is provided by the preceding result. Another is that in power series, such as

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!},$

when x = 0, then the term in which n = 0 has the correct value only if 00 = 1. Yet another is that in combinatorial problems one must sometimes take 00 to be an empty product.

Undefined forms that are not indeterminate

The expression 1/0 is not commonly regarded as an indeterminate form because there is not an infinite range of values that f/g could approach. Specifically, if f approaches 1 and g approaches 0, then f and g may be chosen so that: (1) f/g approaches +∞, (2) f/g approaches −∞, or (3) the limit fails to exist. In each case the absolute value |f/g| approaches +∞, and so the quotient f/g must diverge, in the sense of the extended real numbers. (In the framework of the real projective line, the limit is the unsigned infinity ∞ in all three cases.) Similarly, any expression of the form a/0, with a ≠ 0 (including a = +∞ and a = −∞), is not an indeterminate since a quotient giving rise to such an expression will always diverge.

0 also is sometimes incorrectly thought to be indeterminate: 0+∞ = 0, and 0−∞ is equivalent to 1/0.

Evaluating indeterminate forms

The indeterminate nature of a limit's form does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated.

For example, the expression x2/x can be simplified to x at any point other than x = 0. Thus, the limit of this expression as x approaches 0 (which depends only on points near 0, not at x = 0 itself) is the limit of x, which is 0. Most of the other examples above can also be evaluated using algebraic simplification.

L'Hôpital's rule is a general method for evaluating the indeterminate forms 0/0 and ∞/∞. This rule states that (under appropriate conditions)

$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} , \!$

where f' and g' are the derivatives of f and g. (Note that this rule does not apply to forms like ∞/0, 1/0, and so on; these forms are not indeterminate) With luck, these derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 00:

$\ln \lim_{x \to c} f(x)^{g(x)} = \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} . \!$

The right-hand side is of the form ∞/∞, so L'Hôpital's rule applies to it. Notice that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it's irrelevant how well-behaved f and g may (or may not) be as long as f is asymptotically positive.

Although L'Hôpital's rule applies both to 0/0 and to ∞/∞, one of these may be better than the other in a particular case (because of the possibilities for algebraic simplification afterwards). You can change between these forms, if necessary, by transforming f/g to (1/g)/(1/f).

List of indeterminate forms

The following table lists the indeterminate forms for the standard arithmetic operations and the transformations for applying l'Hôpital's rule.

Indeterminate form Conditions Transformation to 0/0 Transformation to ∞/∞
0/0 $\lim_{x \to c} f(x) = 0,\ \lim_{x \to c} g(x) = 0 \!$
$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)} \!$
∞/∞ $\lim_{x \to c} f(x) = \infty,\ \lim_{x \to c} g(x) = \infty \!$ $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)} \!$
0 × ∞ $\lim_{x \to c} f(x) = 0,\ \lim_{x \to c} g(x) = \infty \!$ $\lim_{x \to c} f(x)g(x) = \lim_{x \to c} \frac{f(x)}{1/g(x)} \!$ $\lim_{x \to c} f(x)g(x) = \lim_{x \to c} \frac{g(x)}{1/f(x)} \!$
1 $\lim_{x \to c} f(x) = 1,\ \lim_{x \to c} g(x) = \infty \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \!$
00 $\lim_{x \to c} f(x) = 0^+, \lim_{x \to c} g(x) = 0 \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \!$
0 $\lim_{x \to c} f(x) = \infty,\ \lim_{x \to c} g(x) = 0 \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \!$ $\lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \!$
∞ − ∞ $\lim_{x \to c} f(x) = \infty,\ \lim_{x \to c} g(x) = \infty \!$ $\lim_{x \to c} (f(x) - g(x)) = \lim_{x \to c} \frac{1/g(x) - 1/f(x)}{1/(f(x)g(x))} \!$ $\lim_{x \to c} (f(x) - g(x)) = \ln \lim_{x \to c} \frac{e^{f(x)}}{e^{g(x)}} \!$