Limit

The value that a sequence or function “approaches” while the index or the input approaches a particular value is described as the concept of limit in mathematics. In order to define the derivatives, continuity and integrals, the limits are found to be an essential tool in the mathematical analysis especially in the field of calculus.

The limit of a sequences concept has been more generalized to form the limit of a topological net whereby it becomes more closely related to the limit and especially the direct limit in the category theory.

"lim" is the abbreviated form of the limit which is used in the formulas lim(a_n) = a or represented by the right arrow (→) as in a_n → a.

Whereby if c is the real number f(x) is the real-valued function, then the expression will go as

f(x) = L

This formula means that by making x significantly close to c, f(x) could be made to close to L. There by the statement “the limit of f of x, as x approaches c, is L" is expressed.

For example, if

f(x) = x² - 1 /x - 1

then f(1) is not defined, yet as x approaches 1, f(x) approaches 2:

There by f(x) can be made arbitrarily close to the limit of 2 just by making x sufficiently close to 1.

The definition for the limit of a certain function has been formalized by Karl Weierstrass. This definition has became known to be the (ε, δ)-definition of limit in the nineteenth century.

There are also functions which have limits infinity in addition to having limits at finite values. For instance,

f(x) =2x - 1 /x

For checking the convergene of p the first step checks whether p is a fixed point.

• f(100) = 1.9900
• f(1000) = 1.9990
• f(10000) = 1.9999
• Since x becomes extremely large, then there is possibility for the value of f(x) to be made close to 2 with the value of f(x) approaching 2. In mathematical notation, this case of the limit of f(X) where the x approaches infinite is equivalent to 2 is formed as
• f(x) = 2

Fixed point and convergence:

The formal definition for convergence has been formed where  p_n as n goes from 0 to  is a sequence which converges to the fixed point p, with   for all n. If there exist constants λ and α then,

Questions:

• What is limit mathematics?
• What is definite of convergence?