Approximate limit

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

A function f on $\mathbb {R} ^{k}$ 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.

Properties

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 $\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x).$

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.)

{\begin{aligned}\lim _{x\rightarrow x_{0}}\operatorname {ap} \ a\cdot f(x)&=a\cdot \lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)\\\lim _{x\rightarrow x_{0}}\operatorname {ap} \ (f(x)+g(x))&=\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)+\lim _{x\rightarrow x_{0}}\operatorname {ap} \ g(x)\\\lim _{x\rightarrow x_{0}}\operatorname {ap} \ (f(x)-g(x))&=\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)-\lim _{x\rightarrow x_{0}}\operatorname {ap} \ g(x)\\\lim _{x\rightarrow x_{0}}\operatorname {ap} \ (f(x)\cdot g(x))&=\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)\cdot \lim _{x\rightarrow x_{0}}\operatorname {ap} \ g(x)\\\lim _{x\rightarrow x_{0}}\operatorname {ap} \ (f(x)/g(x))&=\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)/\lim _{x\rightarrow x_{0}}\operatorname {ap} \ g(x)\end{aligned}}

Approximate continuity and differentiability

If

\lim _{x\rightarrow x_{0}}\operatorname {ap} \ f(x)=f(x_{0})

then f is said to be approximately continuous at x₀. If f is function of only one real variable and the difference quotient

{\frac {f(x_{0}+h)-f(x_{0})}{h}}

has an approximate limit as h approaches zero we say that f has an approximate derivative at x₀. 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.

External links

References

Bruckner, Andrew (1994), Differentiation of real functions (Second ed.), AMS Bookstore, ISBN 0-8218-6990-6
Tolstov, G.P. (2001) [1994], "Approximate limit", Encyclopedia of Mathematics, EMS Press