Zero Divided By Zero

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. Unlike "0/0", "1/0" is not an indeterminate form.

It means also that the Ratio of two functions ''f'' and ''g'' that approach zero can approach any member of a range of well-defined values, depending on which functions ''f'' and ''g'' are. Whether such a value exists, and what it might be, depends on ''how'' the functions approach zero.

In more formal language, the fact that the Functions ''f''(''x'') and ''g''(''x'') both approach 0 as ''x'' approaches some limit ''c'', is not enough information to evaluate the Limit

:

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

That limit could be any number or plus or minus infinity, or might not exist, depending on what the functions ''f'' and ''g'' are.

By contrast "1/0" is not an indeterminate form because there is no range of different values that ''f''/''g'' could approach if ''f'' approaches 1 and ''g'' approaches 0. (The absolute value of the ratio will always approach

+\infty

and so is not considered indeterminate.)

EXAMPLES ON 0/0

For example,

:

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

and

:

\lim_{x
ightarrow 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 Limit s 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.

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Zero Divided By Zero