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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 () can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression). Cases 2–4 are handled similarly. See the examples below.
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 .
The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in ∫ 1 x d x = ln | x | + C {\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C} there is a singularity at 0 and the antiderivative becomes infinite there.
A (real) polynomial is an expression of the form a 0 x 0 + ⋅⋅⋅ + a n x n, where x is an indeterminate, and the coefficients a i are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring R[x].
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L'Hôpital's rule - a method in calculus for evaluating indeterminate forms; Indeterminate form - a mathematical expression for which many assignments exist; NaN - the IEEE-754 expression indicating that the result of a calculation is not a number; Primitive notion - a concept that is not defined in terms of previously-defined concepts