In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables.
A function f on has an approximate limit y at a point x if there exists a set F that has density 1 at the point such that if x<sub>n</sub> is a sequence in F that converges towards x then f(x<sub>n</sub>) converges towards y.
The approximate limit of a function, if it exists, is unique. If f has an ordinary limit at x then it also has an approximate limit with the same value.
We denote the approximate limit of f at x<sub>0</sub> by
Many of the properties of the ordinary limit are also true for the approximate limit.
In particular, if a is a scalar and f and g are functions, the following equations are true if values on the right-hand side are well-defined (that is the approximate limits exist and in the last equation the approximate limit of g is non-zero.)
If
then f is said to be approximately continuous at x<sub>0</sub>. If f is function of only one real variable and the difference quotient
has an approximate limit as h approaches zero we say that f has an approximate derivative at x<sub>0</sub>. It turns out that approximate differentiability implies approximate continuity, in perfect analogy with ordinary continuity and differentiability.
It also turns out that the usual rules for the derivative of a sum, difference, product and quotient have straightforward generalizations to the approximate derivative. There is no generalization of the chain rule that is true in general however.