In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function. For example,
<math display=block>\begin{align}
\lim_{x \to c} \bigl(f(x) + g(x)\bigr) &= \lim_{x \to c} f(x) + \lim_{x \to c} g(x), \\[3mu]
\lim_{x \to c} \bigl(f(x)g(x)\bigr) &= \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x),
\end{align}</math>
and likewise for other arithmetic operations; this is sometimes called the algebraic limit theorem. However, certain combinations of particular limiting values cannot be computed in this way, and knowing the limit of each function separately does not suffice to determine the limit of the combination. In these particular situations, the limit is said to take an indeterminate form, described by one of the informal expressions
<math display=block>\frac 00,~ \frac{\infty}{\infty},~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ or } \infty^0,</math>
among a wide variety of uncommon others, where each expression stands for the limit of a function constructed by an arithmetical combination of two functions whose limits respectively tend to or as indicated.
A limit taking one of these indeterminate forms might tend to zero, might tend to any finite value, might tend to infinity, or might diverge, depending on the specific functions involved. A limit which unambiguously tends to infinity, for instance <math display=inline>\lim_{x \to 0} 1/x^2 = \infty,</math> is not considered indeterminate. 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 <math>0/0</math>. For example, as <math>x</math> approaches <math>0,</math> the ratios <math>x/x^3</math>, <math>x/x</math>, and <math>x^2/x</math> go to <math>\infty</math>, <math>1</math>, and <math>0</math> respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is <math>0/0</math>, which is indeterminate. In this sense, <math>0/0</math> can take on the values <math>0</math>, <math>1</math>, or <math>\infty</math>, 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, <math> x \sin(1/x) / x</math>.
So the fact that two functions <math>f(x)</math> and <math>g(x)</math> converge to <math>0</math> as <math>x</math> approaches some limit point <math>c</math> is insufficient to determine the limit
<math display="block"> \lim_{x \to c} \frac{f(x)}{g(x)} .</math>
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 <math>0^0</math>. Whether this expression is left undefined, or is defined to equal <math>1</math>, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that <math>0^\infty</math> and other expressions involving infinity are not indeterminate forms.
Some examples and non-examples
Indeterminate form 0/0
<gallery class=skin-invert-image>
File:Indeterminate form - x over x.gif|alt=Graph showing a horizontal line at y = 1|Fig. 1: =
File:Indeterminate form - x2 over x.gif|alt=Graph showing a straight line passing from lower left to upper right through the origin with a slope of 1|Fig. 2: =
File:Indeterminate form - sin x over x close.gif|alt=Graph showing a curve that oscillates across the x axis with increasing magnitude towards the y axis, intersecting it at y = 1|Fig. 3: =
File:Indeterminate form - complicated.gif|alt=Graph showing an increasing curve with decreasing slope, vanishing rapidly towards the origin|Fig. 4: = (for = 49)
File:Indeterminate form - 2x over x.gif|alt=Graph showing a horizontal line at y = 2|Fig. 5: = where = 2
File:Indeterminate form - x over x3.gif|alt=Graph asymptotically approaching infinity from both sides of the y axis, and asymptomatically approaching the x axis away from it|Fig. 6: =
</gallery>
The indeterminate form <math>0/0</math> is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.
As mentioned above,
while
{x} = 0, \qquad </math> (see fig. 2)
This is enough to show that <math>0/0</math> is an indeterminate form. Other examples with this indeterminate form include
and
Direct substitution of the number that <math>x</math> approaches into any of these expressions shows that these are examples correspond to the indeterminate form <math>0/0</math>, but these limits can assume many different values. Any desired value <math>a</math> can be obtained for this indeterminate form as follows:
The value <math>\infty</math> can also be obtained (in the sense of divergence to infinity):
Indeterminate form 0<sup>0</sup>
The following limits illustrate that the expression <math>0^0</math> is an indeterminate form:
<math display="block"> \begin{align}
\lim_{x \to 0^+} x^0 &= 1, \\
\lim_{x \to 0^+} 0^x &= 0.
\end{align} </math>
Thus, in general, knowing that <math>\textstyle\lim_{x \to c} f(x) \;=\; 0</math> and <math>\textstyle\lim_{x \to c} g(x) \;=\; 0</math> is not sufficient to evaluate the limit
<math display="block"> \lim_{x \to c} f(x)^{g(x)}. </math>
If the functions <math>f</math> and <math>g</math> are analytic at <math>c</math>, and <math>f</math> is positive for <math>x</math> sufficiently close (but not equal) to <math>c</math>, then the limit of <math>f(x)^{g(x)}</math> will be <math>1</math>. Otherwise, use the transformation in the table below to evaluate the limit.
Expressions that are not indeterminate forms
The expression <math>1/0</math> is not commonly regarded as an indeterminate form, because if the limit of <math>f(x)/g(x)</math> as <math>g(x) \rarr 0</math> exists<!--without loss of generality as to the behaviour of f(x) since we can always force it to approach 1 by introducing a constant factor--> then there is no ambiguity as to its value, as it always diverges. Specifically, with the constraint that <math>f(x)</math> approaches <math>1</math> and <math>g(x)</math> approaches <math>0,</math> we may choose <math>f</math> and <math>g</math> so that:
- <math>f(x)/g(x)</math> approaches <math>+\infty</math>
- <math>f(x)/g(x)</math> approaches <math>-\infty</math>
- The limit fails to exist.
In each case the absolute value <math>|f(x)/g(x)|</math> approaches <math>+\infty</math>, and so the quotient <math>f(x)/g(x)</math> must diverge, in the sense of the extended real numbers (in the framework of the projectively extended real line, the limit is the unsigned infinity <math>\infty</math> in all three cases). Similarly, any expression of the form <math>a/0</math> with <math>a\ne0</math> (including <math>a=+\infty</math> and <math>a=-\infty</math>) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.
The expression <math>0^\infty</math> is not an indeterminate form. The expression <math>0^{+\infty}</math> obtained from considering <math>\lim_{x \to c} f(x)^{g(x)}</math> gives the limit <math>0,</math> provided that <math>f(x)</math> remains nonnegative as <math>x</math> approaches <math>c</math>. The expression <math>0^{-\infty}</math> is similarly equivalent to <math>1/0</math>; if <math>f(x) > 0</math> as <math>x</math> approaches <math>c</math>, the limit comes out as <math>+\infty</math>.
To see why, let <math>L = \lim_{x \to c} f(x)^{g(x)},</math> where <math> \lim_{x \to c} {f(x)}=0,</math> and <math> \lim_{x \to c} {g(x)}=\infty.</math> By taking the natural logarithm of both sides and using <math> \lim_{x \to c} \ln{f(x)}=-\infty,</math> we get that <math>\ln L = \lim_{x \to c} ({g(x)}\times\ln{f(x)})=\infty\times{-\infty}=-\infty,</math> which means that <math>L = {e}^{-\infty}=0.</math>
Evaluating indeterminate forms
The adjective indeterminate 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.
Equivalent infinitesimal
When two variables <math>\alpha</math> and <math>\beta</math> converge to zero at the same limit point and <math>\textstyle \lim \frac{\beta}{\alpha} = 1</math>, they are called equivalent infinitesimal (equiv. <math>\alpha \sim \beta</math>).
Moreover, if variables <math>\alpha'</math> and <math>\beta'</math> are such that <math>\alpha \sim \alpha'</math> and <math>\beta \sim \beta'</math>, then:
<math display="block">\lim \frac{\beta}{\alpha} = \lim \frac{\beta'}{\alpha'}</math>
Here is a brief proof:
Suppose there are two equivalent infinitesimals <math>\alpha \sim \alpha'</math> and <math>\beta \sim \beta'</math>.
<math display=block>\lim \frac{\beta}{\alpha} = \lim \frac{\beta \beta' \alpha'}{\beta' \alpha' \alpha} = \lim \frac{\beta}{\beta'} \lim \frac{\alpha'}{\alpha} \lim \frac{\beta'}{\alpha'} = \lim \frac{\beta'}{\alpha'}</math>
For the evaluation of the indeterminate form <math>0/0</math>, one can make use of the following facts about equivalent infinitesimals (e.g., <math>x\sim\sin x</math> if x becomes closer to zero):
<math display="block">\begin{align}
x &\sim \sin x,\\
x &\sim \arcsin x,\\
x &\sim \sinh x,\\
x &\sim \tan x,\\
x &\sim \arctan x,\\
x &\sim \ln(1 + x),\\
1 - \cos x &\sim \frac{x^2}{2},\\
\cosh x - 1 &\sim \frac{x^2}{2},\\
a^x - 1 &\sim x \ln a,\\
e^x - 1 &\sim x,\\
(1 + x)^a - 1 &\sim ax.
\end{align}</math>
For example:
<math display=block>\begin{align}
\lim_{x \to 0} \frac{1}{x^3} \left[\left(\frac{2+\cos x}{3}\right)^x - 1 \right]
&= \lim_{x \to 0} \frac{e^{x\ln{\frac{2 + \cos x}{3}-1}{x^3} \\
&= \lim_{x \to 0} \frac{1}{x^2} \ln \frac{2+ \cos x}{3} \\
&= \lim_{x \to 0} \frac{1}{x^2} \ln \left(\frac{\cos x -1}{3}+1\right) \\
&= \lim_{x \to 0} \frac{\cos x -1}{3x^2} \\
&= \lim_{x \to 0} -\frac{x^2}{6x^2} \\
&= -\frac{1}{6}
\end{align}</math>
In the 2nd equality, <math>e^y - 1 \sim y</math> where <math>y = x\ln{2+\cos x \over 3}</math> as y become closer to 0 is used, and <math>y \sim \ln {(1+y)}</math> where <math>y = {e^{g(x) \! </math>
|-
|<math>0^0</math>
|<math> \lim_{x \to c} f(x) = 0^+, \lim_{x \to c} g(x) = 0 \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \! </math>
|-
|<math>1^\infty</math>
|<math> \lim_{x \to c} f(x) = 1,\ \lim_{x \to c} g(x) = \infty \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \! </math>
|-
|<math>\infty^0</math>
|<math> \lim_{x \to c} f(x) = \infty,\ \lim_{x \to c} g(x) = 0 \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{g(x)}{1/\ln f(x)} \! </math>
|<math> \lim_{x \to c} f(x)^{g(x)} = \exp \lim_{x \to c} \frac{\ln f(x)}{1/g(x)} \! </math>
|}
See also
- Defined and undefined
- Division by zero
- Extended real number line
- Indeterminate equation
- Indeterminate system
- Indeterminate (variable)
- L'Hôpital's rule