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 ${\displaystyle \mathbb {R} ^{k}}$ 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.

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

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.)

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

Approximate continuity and differentiability

If

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

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

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

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

• Approximate continuity at Encyclopedia of Mathematics
• Approximate derivative at Encyclopedia of Mathematics
• Approximate differentiability at Encyclopedia of Mathematics

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