Indeterminate form: Difference between revisions

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and likewise for other arithmetic operations; this is sometimes called the [[limit of a function#Properties|algebraic limit theorem]]. However, certain combinations of particular limiting values cannot be computed in this way, and knowing the limit of each function separately does not suffice to determine the limit of the combination. In these particular situations, the limit is said to take an '''indeterminate form''', described by one of the informal expressions

<math display=block>\frac 00,~ \frac{\infty}{\infty},~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ or } \infty^0,</math>

* <math>0 \div 0</math>

* <math> {\infty} \div {\infty} </math>

* <math> 0\times\infty </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> \log_{0}(0) </math>

* <math> \log_{1}(1) </math>

* <math> \log_{\infty}(\infty) </math>

where each expression stands for the limit of a function constructed by an arithmetical combination of two functions whose limits respectively tend to {{tmath|0,}} {{tmath|1,}} or {{tmath|\infty}} as indicated.{{sfnp|Varberg|Purcell|Rigdon|2007|p=423, 429, 430, 431, 432}}