Indeterminate form#Indeterminate form 0/0

{{Redirect|0/0|the symbol|Percent sign|0 divided by 0|Division by zero}}

File:Indeterminate form - x over x.gif|Fig. 1: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}}}

File:Indeterminate form - x2 over x.gif|Fig. 2: {{var|y}} = {{sfrac|{{var|x}}{{sup|2}}|{{var|x}}}}

File:Indeterminate form - sin x over x close.gif|Fig. 3: {{var|y}} = {{sfrac|sin {{var|x}}|{{var|x}}}}

File:Indeterminate form - complicated.gif|Fig. 4: {{var|y}} = {{sfrac|x − 49|{{radic|x}} − 7}} (for {{var|x}} = 49)

File:Indeterminate form - 2x over x.gif|Fig. 5: {{var|y}} = {{sfrac|{{var|a}}{{var|x}}|{{var|x}}}} where {{var|a}} = 2

File:Indeterminate form - x over x3.gif|Fig. 6: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}{{sup|3}}}}

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

As mentioned above,

{{block indent|$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{x\}\{x\}\; =\; 1,\; \backslash qquad$ (see fig. 1)}}

while

{{block indent|$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{x^\{2\}\}\{x\}\; =\; 0,\; \backslash qquad$ (see fig. 2)}}

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

{{block indent|$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{\backslash sin(x)\}\{x\}\; =\; 1,\; \backslash qquad$ (see fig. 3)}}

and

{{block indent|$\backslash lim\_\{x\; \backslash to\; 49\}\; \backslash frac\{x\; -\; 49\}\{\backslash sqrt\{x\}\backslash ,\; -\; 7\}\; =\; 14,\; \backslash qquad$ (see fig. 4)}}

Direct substitution of the number that $x$ approaches into any of these expressions shows that these are examples correspond to the indeterminate form $0/0$, but these limits can assume many different values. Any desired value $a$ can be obtained for this indeterminate form as follows:

{{block indent|$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{ax\}\{x\}\; =\; a\; .\; \backslash qquad$ (see fig. 5)}}

The value $\backslash infty$ can also be obtained (in the sense of divergence to infinity):

{{block indent|$\backslash lim\_\{x\; \backslash to\; 0\}\; \backslash frac\{x\}\{x^3\}\; =\; \backslash infty\; .\; \backslash qquad$ (see fig. 6)}}

{{main|Zero to the power of zero}}

{{multiple image

| image1 = Indeterminate form - x0.gif

| caption1 = Graph of {{math|1=y = x{{sup|0}}}}

| image2 = Indeterminate form - 0x.gif

| caption2 = Graph of {{math|1=y = 0{{sup|x}}}}

| total_width = 300

| direction = vertical

}}

The following limits illustrate that the expression $0^0$ is an indeterminate form:

$$\backslash begin\{align\}$$

\lim_{x \to 0^+} x^0 &= 1, \\

\lim_{x \to 0^+} 0^x &= 0.

\end{align}

Thus, in general, knowing that $\backslash textstyle\backslash lim\_\{x\; \backslash to\; c\}\; f(x)\; \backslash ;=\backslash ;\; 0$ and $\backslash textstyle\backslash lim\_\{x\; \backslash to\; c\}\; g(x)\; \backslash ;=\backslash ;\; 0$ is not sufficient to evaluate the limit

$$\backslash lim\_\{x\; \backslash to\; c\}\; f(x)^\{g(x)\}.$$

If the functions $f$ and $g$ are analytic at $c$, and $f$ is positive for $x$ sufficiently close (but not equal) to $c$, then the limit of $f(x)^\{g(x)\}$ will be $1$.{{cite journal |doi=10.2307/2689754 |author1=Louis M. Rotando |author2=Henry Korn |title=The indeterminate form 0⁰ |journal=Mathematics Magazine |date=January 1977 |volume=50 |issue=1 |pages=41–42|jstor=2689754 }} Otherwise, use the transformation in the table below to evaluate the limit.

The expression $1/0$ is not commonly regarded as an indeterminate form, because if the limit of $f/g$ exists then there is no ambiguity as to its value, as it always diverges. Specifically, if $f$ approaches $1$ and $g$ approaches $0,$ then $f$ and $g$ may be chosen so that:

1. $f/g$ approaches $+\backslash infty$
2. $f/g$ approaches $-\backslash infty$
3. The limit fails to exist.

In each case the absolute value $|f/g|$ approaches $+\backslash infty$, and so the quotient $f/g$ 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 $\backslash infty$ in all three cases{{Cite web|url=https://www.cut-the-knot.org/blue/GhostCity.shtml|title=Undefined vs Indeterminate in Mathematics|website=www.cut-the-knot.org|access-date=2019-12-02}}). Similarly, any expression of the form $a/0$ with $a\backslash ne0$ (including $a=+\backslash infty$ and $a=-\backslash infty$) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.

The expression $0^\backslash infty$ is not an indeterminate form. The expression $0^\{+\backslash infty\}$ obtained from considering $\backslash lim\_\{x\; \backslash to\; c\}\; f(x)^\{g(x)\}$ gives the limit $0,$ provided that $f(x)$ remains nonnegative as $x$ approaches $c$. The expression $0^\{-\backslash infty\}$ is similarly equivalent to $1/0$; if $f(x)\; >\; 0$ as $x$ approaches $c$, the limit comes out as $+\backslash infty$.

To see why, let $L\; =\; \backslash lim\_\{x\; \backslash to\; c\}\; f(x)^\{g(x)\},$ where $\backslash lim\_\{x\; \backslash to\; c\}\; \{f(x)\}=0,$ and $\backslash lim\_\{x\; \backslash to\; c\}\; \{g(x)\}=\backslash infty.$ By taking the natural logarithm of both sides and using $\backslash lim\_\{x\; \backslash to\; c\}\; \backslash ln\{f(x)\}=-\backslash infty,$ we get that $\backslash ln\; L\; =\; \backslash lim\_\{x\; \backslash to\; c\}\; (\{g(x)\}\backslash times\backslash ln\{f(x)\})=\backslash infty\backslash times\{-\backslash infty\}=-\backslash infty,$ which means that $L\; =\; \{e\}^\{-\backslash infty\}=0.$

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.

When two variables $\backslash alpha$ and $\backslash beta$ converge to zero at the same limit point and $\backslash textstyle\; \backslash lim\; \backslash frac\{\backslash beta\}\{\backslash alpha\}\; =\; 1$, they are called equivalent infinitesimal (equiv. $\backslash alpha\; \backslash sim\; \backslash beta$).

Moreover, if variables $\backslash alpha\text{'}$ and $\backslash beta\text{'}$ are such that $\backslash alpha\; \backslash sim\; \backslash alpha\text{'}$ and $\backslash beta\; \backslash sim\; \backslash beta\text{'}$, then:

{{block indent|$\backslash lim\; \backslash frac\{\backslash beta\}\{\backslash alpha\}\; =\; \backslash lim\; \backslash frac\{\backslash beta\text{'}\}\{\backslash alpha\text{'}\}$}}

Here is a brief proof:

Suppose there are two equivalent infinitesimals $\backslash alpha\; \backslash sim\; \backslash alpha\text{'}$ and $\backslash beta\; \backslash sim\; \backslash beta\text{'}$.

$$\backslash lim\; \backslash frac\{\backslash beta\}\{\backslash alpha\}\; =\; \backslash lim\; \backslash frac\{\backslash beta\; \backslash beta\text{'}\; \backslash alpha\text{'}\}\{\backslash beta\text{'}\; \backslash alpha\text{'}\; \backslash alpha\}\; =\; \backslash lim\; \backslash frac\{\backslash beta\}\{\backslash beta\text{'}\}\; \backslash lim\; \backslash frac\{\backslash alpha\text{'}\}\{\backslash alpha\}\; \backslash lim\; \backslash frac\{\backslash beta\text{'}\}\{\backslash alpha\text{'}\}\; =\; \backslash lim\; \backslash frac\{\backslash beta\text{'}\}\{\backslash alpha\text{'}\}$$

For the evaluation of the indeterminate form $0/0$, one can make use of the following facts about equivalent infinitesimals (e.g., $x\backslash sim\backslash sin\; x$ if x becomes closer to zero):{{Cite web|url=http://www.vaxasoftware.com/doc_eduen/mat/infiequi.pdf|title=Table of equivalent infinitesimals|website=Vaxa Software}}

{{block indent|$x\; \backslash sim\; \backslash sin\; x,$}}

{{block indent|$x\; \backslash sim\; \backslash arcsin\; x,$}}

{{block indent|$x\; \backslash sim\; \backslash sinh\; x,$}}

{{block indent|$x\; \backslash sim\; \backslash tan\; x,$}}

{{block indent|$x\; \backslash sim\; \backslash arctan\; x,$}}

{{block indent|$x\; \backslash sim\; \backslash ln(1\; +\; x),$}}

{{block indent|$1\; -\; \backslash cos\; x\; \backslash sim\; \backslash frac\{x^2\}\{2\},$}}

{{block indent|$\backslash cosh\; x\; -\; 1\; \backslash sim\; \backslash frac\{x^2\}\{2\},$}}

{{block indent|$a^x\; -\; 1\; \backslash sim\; x\; \backslash ln\; a,$}}

{{block indent|$e^x\; -\; 1\backslash sim\; x,$}}

{{block indent|$(1\; +\; x)^a\; -\; 1\; \backslash sim\; ax.$}}

For example:

$$\backslash begin\{align\}$$

\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}

In the 2nd equality, $e^y\; -\; 1\; \backslash sim\; y$ where $y\; =\; x\backslash ln\{2+\backslash cos\; x\; \backslash over\; 3\}$ as y become closer to 0 is used, and $y\; \backslash sim\; \backslash ln\; \{(1+y)\}$ where $y\; =\; \{\{\backslash cos\; x\; -\; 1\}\; \backslash over\; 3\}$ is used in the 4th equality, and $1-\backslash cos\; x\; \backslash sim\; \{x^2\; \backslash over\; 2\}$ is used in the 5th equality.

{{main|L'Hôpital's rule}}

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

{{block indent|$\backslash lim\_\{x\; \backslash to\; c\}\; \backslash frac\{f(x)\}\{g(x)\}\; =\; \backslash lim\_\{x\; \backslash to\; c\}\; \backslash frac\{f\text{'}(x)\}\{g\text{'}(x)\}\; ,$}}

where $f\text{'}$ and $g\text{'}$ are the derivatives of $f$ and $g$. (Note that this rule does not apply to expressions $\backslash infty/0$, $1/0$, and so on, as these expressions are not indeterminate forms.) 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:

{{block indent|$\backslash ln\; \backslash lim\_\{x\; \backslash to\; c\}\; f(x)^\{g(x)\}\; =\; \backslash lim\_\{x\; \backslash to\; c\}\; \backslash frac\{\backslash ln\; f(x)\}\{1/g(x)\}\; .$}}

The right-hand side is of the form $\backslash infty/\backslash infty$, so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved $f$ and $g$ may (or may not) be as long as $f$ is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.)

Although L'Hôpital's rule applies to both $0/0$ and $\backslash infty/\backslash infty$, one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms by transforming $f/g$ to $(1/g)/(1/f)$.

The following table lists the most common indeterminate forms and the transformations for applying l'Hôpital's rule.

• Defined and undefined
• Division by zero
• Extended real number line
• Indeterminate equation
• Indeterminate system
• Indeterminate (variable)
• L'Hôpital's rule

{{reflist}}

• {{cite book

| last1 = Varberg | first1 = Dale E.

| last2 = Purcell | first2 = Edwin J.

| last3 = Rigdon | first3 = Steven E.

| title = Calculus

| year = 2007

| publisher = Pearson Prentice Hall

| edition = 9th

| isbn = 978-0131469686

}}

{{Calculus topics}}

Category:Limits (mathematics)