In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables.

A function f on ${\displaystyle \mathbb {R} ^{k}}$ 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_n is a sequence in F that converges towards x then f(x_n) 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 x₀ by ${\displaystyle \lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x).}$

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 x₀. If f is function of only one real variable and the difference quotient