mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables. A function ''f'' on

\mathbb^k

has an approximate limit ''y'' at a point ''x'' if there exists a set ''F'' that has

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

1 at the point such that if ''x''_''n'' is a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

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 ''x''₀ by

\lim_ \operatorname \ 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
\lim_ \operatorname \ a\cdot f(x) & =a \cdot \lim_ \operatorname \ f(x) \\
\lim_ \operatorname \ (f(x)+g(x)) & = \lim_ \operatorname \ f(x) + \lim_ \operatorname \ g(x) \\
\lim_ \operatorname \ (f(x)-g(x)) & = \lim_ \operatorname \ f(x)-\lim_ \operatorname \  g(x) \\
\lim_ \operatorname \ (f(x)\cdot g(x)) & = \lim_ \operatorname \ f(x) \cdot \lim_ \operatorname \ g(x) \\
\lim_ \operatorname \ (f(x)/g(x)) & = \lim_ \operatorname \ f(x) / \lim_ \operatorname \ g(x)
\end

Approximate continuity and differentiability

If :

\lim_ \operatorname \ f(x) = f(x_0)

then ''f'' is said to be approximately continuous at ''x''₀. If ''f'' is function of only one real variable and the

difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which when taken to the Limit of a function, limit as ''h'' approaches 0 gives the derivative of the Function (mathematics), function ''f''. The ...

\frac

has an approximate limit as ''h'' approaches zero we say that ''f'' has an approximate derivative at ''x''₀. 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 In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...

that is true in general however.

References

* * {{DEFAULTSORT:Approximate Limit Real analysis Limits (mathematics)

Properties

Approximate continuity and differentiability

External links

References