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Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus , it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function.
Here is a basic example involving the exponential function, which involves the indeterminate form 0 / 0 at x = 0: + = (+) = + = This is a more elaborate example involving 0 / 0 . Applying L'Hôpital's rule a single time still results in an indeterminate form.
The previous remarks about indeterminate forms, iterated limits, and the Cauchy principal value also apply here. The function f ( x ) {\displaystyle f(x)} can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression).
This rule uses derivatives to find limits of indeterminate forms 0/0 or ±∞/∞, and only applies to such cases. Other indeterminate forms may be manipulated into this form. Given two functions f(x) and g(x), defined over an open interval I containing the desired limit point c, then if:
The fallacy here arises from the assumption that it is legitimate to cancel 0 like any other number, whereas, in fact, doing so is a form of division by 0. Using algebra , it is possible to disguise a division by zero [ 17 ] to obtain an invalid proof .
In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. [1] For example, the equation a x + b y = c {\displaystyle ax+by=c} is a simple indeterminate equation, as is x 2 = 1 {\displaystyle x^{2}=1} .
In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values). [1] The term can take on several different meanings depending on the context. For example:
A formal power series in an indeterminate is an expression of the form + + + …, where no value is assigned to the symbol . [7] This is similar to the definition of a polynomial, except that an infinite number of the coefficients may be nonzero.