Indeterminate form: Difference between revisions

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Come to think of it, it's better to include some mathematical operations that are indeterminate, these 3 nth roots are also indeterminate. I think it's better and more organized to place operations one under the other.
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* <math> \sqrt[\infty]{0} </math>
* <math> \sqrt[\infty]{\infty} </math>
* <math> \sqrt[(-\infty)]{0} </math>
* <math> \sqrt[(-\infty)]{\infty} </math>
* <math> \sqrt[(-\infty)]{(-\infty)} </math>
* <math> \sqrt[\infty]{(-\infty)} </math>
* <math> 1^ {(-\infty)} </math>
* <math> (-1)^ {(-\infty)} </math>
* <math> (-1)^ {\infty} </math>
* <math> {(-\infty)} \div {(-\infty)} </math>
* <math> {(-\infty)} - {(-\infty)} </math>
* <math> (-\infty)^0 </math>
* <math> 0\times (-\infty) </math>
* <math> {(-\infty)} \times 0 </math>
* <math> {\infty} + {(-\infty)} </math>
* <math> {\infty} \div {(-\infty)} </math>
* <math> {(-\infty)} + {\infty} </math>
* <math> {(-\infty)} \div {\infty} </math>
 
These seven expressions are known as '''indeterminate forms'''. More specifically, such expressions are obtained by naively applying the [[algebraic limit theorem]] to evaluate the limit of the corresponding arithmetic operation of two functions, yet there are examples of pairs of functions that after being operated on converge to 0, converge to another finite value, diverge to infinity or just diverge. This inability to decide what the limit ought to be explains why these forms are regarded as '''indeterminate'''. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity).<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Indeterminate.html|title=Indeterminate|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> The term was originally introduced by [[Cauchy]]'s student [[Moigno]] in the middle of the 19th century.