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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 ...
For a given iterated function :, the plot consists of a diagonal (=) line and a curve representing = (). To plot the behaviour of a value x 0 {\displaystyle x_{0}} , apply the following steps. Find the point on the function curve with an x-coordinate of x 0 {\displaystyle x_{0}} .
The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined. For some starting values, escape ...
The fixed point iteration x n+1 = cos x n with initial value x 1 = −1.. An attracting fixed point of a function f is a fixed point x fix of f with a neighborhood U of "close enough" points around x fix such that for any value of x in U, the fixed-point iteration sequence , (), (()), ((())), … is contained in U and converges to x fix.
The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout". If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x, y point is examined.
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]
The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers. The same notation is also used for other ways of passing to a limit: e.g. x → 0, x ↓ 0, | x | → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.
Cobweb plot of the Gauss map for = and =.This shows an 8-cycle. In mathematics, the Gauss map (also known as Gaussian map [1] or mouse map), is a nonlinear iterated map of the reals into a real interval given by the Gaussian function: