math lessons - Indeterminate form

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In mathematics, a number of the expressions that may be encountered in calculus and occasionally elsewhere are considered to be indeterminate forms, and must be treated as symbolic only, until more careful discussion has taken place. The most common such expression is 0/0, which has no definite meaning, as division by zero is not a meaningful operation in arithmetic.

Discussion

To say that "0/0" is an indeterminate form does not just mean that "0/0" by itself can represent any number, or may represent no number. Those points are true, in a certain sense, but of limited practical significance when stated in those terms.

It means also that the ratio of two functions that approach zero might approach a well-defined value. Whether such a value exists, and what it might be, depends on how the functions approach zero.

In more formal language, if mathematical functions f(x) and g(x) both approach 0 as x approaches some limit c, one may lack sufficient information to evaluate the mathematical limit

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

That limit could be any number, could be infinite or could fail to exist, depending on what the functions f and g are.

Such limits are important in many areas. To use them correctly, precautions on working with indeterminate forms ought to be observed.

If f(x) and g(x) both approach 0 as x approaches some number, or x approaches ∞ or −∞, then

${f(x) \over g(x)}$

can approach any real number or ∞ or −∞, or fail to converge to any point on the extended real number line, depending on the nature of the functions f and g. Similar remarks are true of the other indeterminate forms displayed below.

Examples on 0/0

For example,

$\lim_{x\rightarrow 0}{\sin(x)\over x}=1$

and

$\lim_{x\rightarrow 49}{x-49\over\sqrt{x}\,-7}=14.$

Direct substitution of the number that x approaches into either of these functions leads to the indeterminate form 0/0, but both limits actually exist and are 1 and 14 respectively.

The indeterminate nature of the form does not imply the limit does not exist. In many cases, algebraic elimination, L'Hôpital's rule, infinity tricks, or other methods can be used to simplify the expression so the limit can be more easily evaluated.

List of indeterminate forms

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

Form	Conditions	Transformation
$\lim f(x)/g(x)$	$\lim f(x)=0$ , $\lim g(x)=0$	none needed
$\lim f(x)/g(x)$	$\lim f(x)=\pm\infty$ , $\lim g(x)=\pm\infty$	none needed
$\lim f(x)\cdot g(x)$	$\lim f(x)=0$ , $\lim g(x)=\pm\infty$	$\lim \frac{f(x)}{1/g(x)}$
$\lim f(x)^{g(x)}$	$\lim f(x)=1$ , $\lim g(x)=\infty$	$e^{(\lim \frac{\ln f(x)}{1/g(x)})}$
$\lim f(x)^{g(x)}$	$\lim f(x)=0$ , $\lim g(x)=0$ [1]	$e^{(\lim \frac{\ln f(x)}{1/g(x)})}$
$\lim f(x)^{g(x)}$	$\lim f(x)=\infty$ , $\lim g(x)=0$	$e^{(\lim \frac{\ln f(x)}{1/g(x)})}$
$\lim (f(x)-{g(x))}$	$\lim f(x)=\infty$ , $\lim g(x)=\infty$	$\ln (\lim \frac{e^{f(x)}}{e^{g(x)}})$