APCALCLimits

// **Calculus AB: Limits - Jack McCabe ** // [|Limits] are an application of [|Calculus] used to denote a value as the function approaches a certain value. Limits are [|notated] in a proper format, and serve as a fundamental part of Calculus. With the notation of the limit, one must determine where the function is heading and what value it is approaching in that direction(s).

=toc 1. History= =2. Notation= =3. Applications= =4. Examples= =5. Multimedia=

**//__History of Limits:__//** Limits date back to the [|19th century] when [|French] [|mathematician], [|Bernhard Bolzano] first introduced [|continuous functions]. Bolzano used his [|patented] [|Epilson-delta] applications to give an accurate definition of a limit. This eventually materialized into our [|present day] understanding of limits and Bolzano as well as renowned mathematician, [|Augustin-Louis Cauchy] are credited for laying the foundation of these concepts.

 **// __﻿Notation: __ //** Limits are notated in a distinct way, and the notation reads a certain way which allows us to determine what direction the function is traveling in, what value it is approaching, and what the resulting value is. Below is a definition and example of limit notation.

** Intuitive Definition. ** Let **y = f(x)** be a function. Suppose that **a** and **L** are numbers such that Then we say that **the limit of f(x) as x approaches a is L** and we write __**//Applications of Limits: //**__ Limits are applicable to various real-life situations, and are used by workers in several different employment fields. [|Engineers] and [|accountants] are the most practical users of limits. Limits serve are a fundamental concept of calculus, and while they are important, they are the foundation of [|derivatives] and [|integrals] as well. All together, limits, derivatives, and integrals are used to solve a wide-variety of engineering problems and are applied in certain features of accounting. 
 * whenever **x** is close to **a** but not equal to **a**, **f(x)** is close to **L**;
 * as **x** gets closer and closer to **a** but not equal to **a**, **f(x)** gets closer and closer to **L**; and
 * suppose that **f(x)** can be made as close as we want to **L** by making **x** close to **a** but not equal to **a**.

__//**Examples:**//__ ||  || >> must exist. > > **Problem:** > > **Solution**: >
 * || **Definition.** A function **f** is **continuous at x = a** if and only if
 * If a function **f** is continuous at **x = a**then we must have the following three conditions.
 * 1) **f(a)** is defined; in other words, **a** is in the domain of **f**.
 * 2) The limit
 * 1) The two numbers in 1. and 2., **f(a)** and **L**, must be equal.

**//__Rules of Limits:__//**

**Theorem A.** A [|polynomial] is continuous at each [|real number]. A [|rational function] is continuous at each point of its [|domain].
 * **Theorem B.** Suppose that **f** and **g** are functions which are continuous at the point **x = a** and suppose that **k** is a constant. Then
 * 1) The product **k f** is continuous at **x = a**.
 * 2) The sum **f + g** is continuous at **x = a**.
 * 3) The difference **f - g** is continuous at **x = a**.
 * 4) The product **f g** is continuous at **x = a**.
 * 5) The quotient **f / g** is continuous at **x = a** provided that **g(a)** is not zero. ||


 * **Theorem C.** Suppose that **g** is a function which is continuous at **x = a** and suppose that **f** is a function which is continuous at **x = g(a)** then the composition of **f** and **g** is continuous at **x = a**. ||

**//__Multimedia:__//**

//"The Limit does not Exist!" - [|Mean Girls] (@ 7:10)// media type="youtube" key="QIMSC-RWvF8" height="349" width="560"

media type="youtube" key="AUFUtweYUCo" height="349" width="425"media type="youtube" key="HB8CzZEd4xw" height="349" width="560"

<span style="color: #ff0000; font-family: 'Times New Roman',Times,serif; font-size: 140%;">**__//References://__** "Continuous Functions - 5." //Mathematics Archives WWW Server//. Web. 06 June 2011. [].

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"File:Augustin-Louis Cauchy 1901.jpg." //Wikipedia, the Free Encyclopedia//. Web. 06 June 2011. .

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"Limit (mathematics)." //Wikipedia, the Free Encyclopedia//. Web. 06 June 2011. [].

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//YouTube - Broadcast Yourself.// Web. 06 June 2011. [].

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//YouTube - Broadcast Yourself.// Web. 06 June 2011. <http://www.youtube.com/watch?v=QIMSC-RWvF8>.