Search results
Results from the WOW.Com Content Network
In this case, a single limit does not exist because the one-sided limits, and + exist and are finite, but are not equal: since, +, the limit does not exist. Then, x 0 {\displaystyle x_{0}} is called a jump discontinuity , step discontinuity , or discontinuity of the first kind .
In arithmetic, and therefore algebra, division by zero is undefined. [7] Use of a division by zero in an arithmetical calculation or proof, can produce absurd or meaningless results. Assuming that division by zero exists, can produce inconsistent logical results, such as the following fallacious "proof" that one is equal to two [8]:
However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits. An example is the expression . Whether this expression is left undefined, or is defined to equal , depends on the field of application and may vary between authors.
This process does not guarantee success; a limit might fail to exist, or might be infinite. For example, over the bounded interval from 0 to 1 the integral of 1/x does not converge; and over the unbounded interval from 1 to ∞ the integral of 1/ √ x does not converge. The improper integral
A critical point of a function of a single real variable, f (x), is a value x 0 in the domain of f where f is not differentiable or its derivative is 0 (i.e. ′ =). [2] A critical value is the image under f of a critical point.
In computing, NaN (/ n æ n /), standing for Not a Number, is a particular value of a numeric data type (often a floating-point number) which is undefined as a number, such as the result of 0/0. Systematic use of NaNs was introduced by the IEEE 754 floating-point standard in 1985, along with the representation of other non-finite quantities ...
If is a set, then there exists precisely one function from to , the empty function. As a result, the empty set is the unique initial object of the category of sets and functions. The empty set can be turned into a topological space , called the empty space, in just one way: by defining the empty set to be open .
This is because is not, and does not designate, a number. The distinction is subtle, since a polynomial in X {\displaystyle X} can be changed to a function in x {\displaystyle x} by substitution. But the distinction is important because information may be lost when this substitution is made.