Content deleted Content added
better to just include classic indeterminate forms, and not to try to be exhaustive in a list of examples |
No edit summary Tag: Reverted |
||
Line 2:
In [[calculus]] and other branches of [[mathematical analysis]], when the [[limit (mathematics)|limit]] of the sum, difference, product, quotient or power of two functions is taken, it may often be possible to simply add, subtract, multiply, divide or exponentiate the corresponding limits of these two functions respectively. However, there are occasions where it is unclear what the sum, difference, product, quotient, or power of these two limits ought to be. For example, it is unclear what the following expressions ought to evaluate to:<ref name=":1" />
* <math>0 \div 0</math>
* <math> {\infty} \div {\infty} </math>
* <math> 0\times\infty </math>
* <math> {\infty} \times 0 </math>
* <math> \infty - \infty </math>
* <math> 0^0 </math>
* <math> 1^\infty </math>
* <math> \infty^0 </math>
* <math> \sqrt[0]{1} </math>
* <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.
| |||