Approximate limit explained

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

A function f on

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 → 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{align} \lim _{x → x _{0}} \operatorname{ap} a ⋅ f(x) & = a ⋅ \lim _{x → x _{0}} \operatorname{ap} f(x) \\ \lim _{x → x _{0}} \operatorname{ap} (f(x) + g(x)) & = \lim _{x → x _{0}} \operatorname{ap} f(x) + \lim _{x → x _{0}} \operatorname{ap} g(x) \\ \lim _{x → x _{0}} \operatorname{ap} (f(x) - g(x)) & = \lim _{x → x _{0}} \operatorname{ap} f(x) - \lim _{x → x _{0}} \operatorname{ap} g(x) \\ \lim _{x → x _{0}} \operatorname{ap} (f(x) ⋅ g(x)) & = \lim _{x → x _{0}} \operatorname{ap} f(x) ⋅ \lim _{x → x _{0}} \operatorname{ap} g(x) \\ \lim _{x → x _{0}} \operatorname{ap} (f(x)/g(x)) & = \lim _{x → x _{0}} \operatorname{ap} f(x) / \lim _{x → x _{0}} \operatorname{ap} g(x) \end{align}

Approximate continuity and differentiability

If

\lim _{x → 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(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