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In multivariable calculus, an iterated limit is a limit of a sequence or a limit of a function in the form , = (,), (,) = ((,)),or other similar forms. An iterated limit is only defined for an expression whose value depends on at least two variables. To evaluate such a limit, one takes the limiting process as one of the two variables approaches some number, getting an expression whose value ...
If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions. For example, f −1 ( x ) is the normal inverse of f , while f −2 ( x ) is the inverse composed with itself, i.e. f −2 ( x ) = f −1 ( f −1 ( x )) .
If f has an incomplete domain, it is possible for Newton's method to send the iterates outside of the domain, so that it is impossible to continue the iteration. [19] For example, the natural logarithm function f(x) = ln x has a root at 1, and is defined only for positive x. Newton's iteration in this case is given by
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is zero, even if the function is undefined (or has another value) at that point (for example at x = 0 for the function given by f (x) = abs(x) − x 2 when x ≠ 0 and by f (0) = 5, starting with the interval [-0.5, 3.0]).
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points and for converging rapidly to these fixed points. The behavior of general root-finding algorithms is studied in numerical analysis .
In general, any infinite series is the limit of its partial sums. For example, an analytic function is the limit of its Taylor series, within its radius of convergence. = =. This is known as the harmonic series. [6]