Final+Version

Limits of functions**
 * [[image:3dpr1noel.gif]]

I. __Definition__

The limit of a function, describes the behavior of a function near a particular input, in other words the limit of f(x) is equal to "L" when "x" approaches "b", if you can find a "x" sufficiently close to "b" such that the value of f(x) is as close to "L" as desired.



A formal definition is, given a function f(x), it is said that it has limit equal to "L" when "x" tends to "b" if for every épsilon positive there is a delta (a real number) also positive such that for the module of the distance from "x" up to "b" between 0 and delta, the module of the difference of f(x) minus L is minor that epsilon.

II. __Description__

Properties:
 * If the limit of a function exists that limit is unique.


 * Limit of the sum/subtraction is the sum/subtraction of the limits, it means that the limit when "x" approaches "b" of f(X) plus or minus g(x) it is equal to the limit when "x" approaches "b" of f(X) plus or minus the limit when "x" approaches "b" of g(X).






 * Limit of the multiplication is the multiplication of the limits, it means that the limit when "x" approaches "b" of f(X) times g(x) it is equal to the limit when "x" approaches "b" of f(X) times the limit when "x" approaches "b" of g(X).


 * If g(x) is different of zero, then the limit of the division is the division of the limits, it means that the limit when "x" approaches "b" of f(X) divided by g(x) it is equal to the limit when "x" approaches "b" of f(X) divided by the limit when "x" approaches "b" of g(X).


 * For "k" constant, the limit when "x" approaches"b" of f(X) times "k" it is equal to "k" times the limit when "x" approaches "b" of f(X).


 * The limit when "x" approaches "b" of the nth root of f (x) is equal to the nth root of the limit when "x" approaches "b" of f (x).

III. Classification

The limit of a function can be classified as follows:

1. Limit of a function at a point: Given a function whose domain is contained in the real number set and a point of accumulation "A" that belongs to the domain, there is a fuction which limit is "L" in "a" for "x" approaching "a" and the function that is close to "L".

Theorem: the limit of a function at a point exists if and only if, there are lateral limits and these are the same.

2. Lateral limits of a function: Will evaluate the limit of the function at a point "a" given but on the left (-) and right (+), as follows: a. From the left: the same as the limit of a function at a point only that the "x" is going to approach be closer to "a" for values less than "a". b. From the right: the same as the limit of a function at a point only that the "x" is going to approach be closer to "a" for values greater than "a".

3. Limits to infinity of a function: Lateral limits are evaluated for function at infinity, as follows: a. From the left: the same as the limit of a function at a point only that the "x" is approaching to "minus infinity". b. From the right: the same as the limit of a function at a point only that the "x" is approaching to "plus infinity".

Moreover, if the function is a polynomial fraction eg (A(x^k))/(B(x^m)). Three cases occur: I. If k=m then the limit of the function when "x" is approaching to infinity is equal to A/B. II. If km then the limit of the function when "x" is approaching to infinity is equal to infinity.

IV. Comparison & Contrast

Lateral limits and the limits to infinity are similar since both evaluate a function at a point, and whatever the point is, this will be evaluated by both sides, left and right. The difference is that in the lateral limits are evaluated at a point "a" anyone, while the limits to infinity are always evaluated at "minus infinity" for the left and "more infinite" on the right.