Is the Universe Entirely Mathematical? Feat. Max Tegmark. Calculus Capers. The Limit Concept and its Definition. We use MathJax Underlying all of calculus is the idea of a limit.
In this section, we will explore what a limit is. Functions with Holes Through most of your earlier work in mathematics, you worked with some very nice functions. But there are quite a few functions which exhibit some rather unusual behaviors. So how do we describe the location of the hole? Closing in on the Hole Let us suppose now that we have a generic function y=f(x) with a generic point (x,f(x)), and with a hole located at the point (c,L). To describe the behavior of the function near the hole, we will speak in terms of the point as it approaches the hole.
When the value of x approaches c, then the value of f(x) approaches L. That statement roughly describes what is happening, and our notation for this statement, \lim\limits_{x\to c} f(x)=L, reflects these very words. Variations on the Limit Definition. We use MathJax Several different varieties of limits exist.
On this page, we collect them all. The Original (Two-Sided) Limit The expression \lim\limits_{x\to c} f(x)=L means: For every \epsilon >0, there exists a \delta >0, such that for every x, the expression 0 < |x-c| < \delta will imply |f(x)-L| < \epsilon. This is the basic two-sided limit that we described on a previous page. One-Sided Limits These forms allow us to describe the behavior of a function from one side only. One-sided limits are critically important when the domain of the function is only on one side of the limit point, as in the function f(x)=\sqrt{x}. Limits at Infinity Limits at infinity are very useful when testing and describing horizontal asymptotes.
Infinite Limits Infinite limits are useful when testing and describing vertical asymptotes. One-Sided Infinite Limits Quite often, a function will have a vertical asymptote, but head off in different directions on the two sides of that asymptote.
GeoGebra. GeoGebraTube. Geometry. Organic Patterns. Fractal. Figure 1a.
The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined. Figure 1b. The same fractal magnified six times. Figure 1c. Figure 1d. Fractals are distinguished from regular geometric figures by their fractal dimensional scaling. As mathematical equations, fractals are usually nowhere differentiable.[2][5][8] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.[7]:48[2]:15 There is some disagreement amongst authorities about how the concept of a fractal should be formally defined.
Introduction[edit] The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. History[edit] Figure 2. Mandelbrot set. Initial image of a Mandelbrot set zoom sequence with a continuously colored environment Mandelbrot animation based on a static number of iterations per pixel remains bounded.[1] That is, a complex number c is part of the Mandelbrot set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0,..., which is bounded, and so −1 belongs to the Mandelbrot set. Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. History[edit] The first picture of the Mandelbrot set, by Robert W. Formal definition[edit] Quick Guide to the Mandelbrot Set. Visitors since May 25 1998 according to my LE FastCounter.
LinkExchange Member About This Page This guide assumes your primary interest in the M-set is as an explorer, and that you have a fractal-drawing program such as FractInt and know how to zoom with it. (If you are looking for a fractal-generating program click here. If you do not know how to bring up an image of the M-set and zoom into it read the program's documentation.) Misiurewicz point. Mathematical notation[edit]
Info. « hd fractals.