Indeterminate form: Difference between revisions

Revision as of 21:59, 20 September 2004

Revision as of 22:49, 20 September 2004

Line 3:

:<math>{0 \over 0}</math>

which has no definite meaning, considering that [[division by zero]] is not a meaningful operation in [[arithmetic]]. If one knows that ''f''(''x'') and ''g''(''x'') both approach 0 as ''x'' approaches any particular limit ''c'', one lacks sufficient information to evaluate the limit

<math>\lim_{x\to c}{f(x) \over g(x)};</math>

Further examples are

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

all==List of which are firstly indeterminate forms. ==

All of the following are indeterminate forms.

:<math>\infty/\infty</math>

Line 19 ⟶ 25:

:<math>\infty-\infty,</math>

==More on 0/0==

all of which are firstly indeterminate forms.

If ''f''(''x'') and ''g''(''x'') both approach 0 as ''x'' approaches some number, or ''x'' approaches &infin; or &minus;&infin;, then

Line 35 ⟶ 41:

Direct substitution of the number that ''x'' approaches into either of these functions leads to the indeterminate form 0/0, but both [[limit of a function|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]], or other methods can be used to simplify the expression so the limit can be more easily evaluated.

==Naive arguments to give indeterminate forms a meaning==

There are many naive reasons which may be given for considering indeterminate forms to have some definite meaning (for example):

Line 45 ⟶ 52:

The above are true statements if they are qualified by exceptions. Below are the correct versions of these statements.

*Any non zerononzero number divided by itself is 1.

*Any non zerononzero number to the power of 0 is 1.

The problem with <math>\infty</math> runs deeper. The symbol <math>\infty</math> isdoes not meant to represent a number. It represents a [[limit]] only. As such the following statements are entirely meaningless:

*Anything multiplied by <math>\infty</math> is <math>\infty,</math> Hencehence <math>0 \cdot \infty =\infty.</math>

*Anything dividedmultiplied by <math>\infty</math>0 is 0, Hencehence <math>0\infty /cdot\infty =0.</math>

*Anything divided by <math>\infty</math> is 0 Hence <math>\infty /\infty =0.</math>

However there(There are defined concepts such as the [[surreal number]]s and the [[ordinals]] where operations on infinite algebraobjects hasare beenwell defined.)

==Logical circularity==

Line 70 ⟶ 78:

in the evaluation of the limit, then one's reasoning is circular and therefore fallacious.

[[Category:Mathematical analysis]]