Indeterminate form: Difference between revisions

Revision as of 10:45, 10 December 2003

Revision as of 02:03, 29 January 2004

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In [[calculus]], the expressions

:<math>{0/0\over0}</math>

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

:<math>0\cdot\infty</math>

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:<math>\infty^0</math>

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

[''others?'']

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

:<math>{f(x)/ \over g(x)}</math>

can approach any real number or &infin; or &minus; &infin;, or fail to converge to any point on the [[extended real number line]], depending on which functions ''f'' and ''g'' are; similar remarks are true of the other indeterminate forms displayed above. For example,

:<math>\lim_{x\rightarrow 0}{\sin(x)\over x}=1</math>

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:<math>\lim_{x\rightarrow 49}{x-49\over\sqrt{x}\,-7}=14.</math>

Direct substitution of the number that ''x'' approaches into either of these functions leads to the indeterminate form 0/0, but both limits[[limit of a function|limit]]s actually exist and are 1 and 14 respectively.

The indeterminate form does not imply the limit does not exist. Algebraic elimination or applying [[L'Hopital's rule]] can be used to simplify the expression so the limit can be more easily and actually evaluated.

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