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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.
Indeterminate (variable), a symbol that is treated as a variable; Indeterminate system, a system of simultaneous equations that has more than one solution; Indeterminate equation, an equation that has more than one solution; Indeterminate form, an algebraic expression with certain limiting behaviour in mathematical analysis
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.
A fundamental property of an indeterminate is that it can be substituted with any mathematical expressions to which the same operations apply as the operations applied to the indeterminate. Some authors of abstract algebra textbooks define an indeterminate over a ring R as an element of a larger ring that is transcendental over R.
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} .
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].
An indeterminate system by definition is consistent, in the sense of having at least one solution. [3] For a system of linear equations, the number of equations in an indeterminate system could be the same as the number of unknowns, less than the number of unknowns (an underdetermined system ), or greater than the number of unknowns (an ...
The powr function defines all three indeterminate forms as invalid operations and so returns NaN. Real operations with complex results, for example: The square root of a negative number. The logarithm of a negative number. The inverse sine or inverse cosine of a number that is less than −1 or greater than 1.