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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 mathematics, the term undefined refers to a value, function, or other expression that cannot be assigned a meaning within a specific formal system. [1]Attempting to assign or use an undefined value within a particular formal system, may produce contradictory or meaningless results within that system.
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.
If the one-sided limits exist at p, but are unequal, then there is no limit at p (i.e., the limit at p does not exist). If either one-sided limit does not exist at p, then the limit at p also does not exist. A formal definition is as follows. The limit of f as x approaches p from above is L if:
The limits in this case are not infinite, but rather undefined: there is no value that () settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity . The possible cases at a given value c {\displaystyle c} for the argument are as follows.
This is an accepted version of this page This is the latest accepted revision, reviewed on 9 January 2025. Undefined may refer to: Mathematics Undefined (mathematics), with several related meanings Indeterminate form, in calculus Computing Undefined behavior, computer code whose behavior is not specified under certain conditions Undefined value, a condition where an expression does not have a ...
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.
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