LimitsAnalytically2

**Evaluating Limits of //F//(x) Analytically!** Created By Sade King, Tranise Ross, and Ryan Giraud (:

First, what are limits?! Lets look at limits in a geometric example. The polygons in these circles, if we increase the number of sides of the polygon will the perimeter be greater than the area? lets look at these examples:







__**To get a simpler and a better understanding on limits: [|Click here!]**__

This graph is both horizontal and vertical asymptot. An Asymptot is a line that continuously approach a given curve but does not meet any defintie distance. lets look at the ** blue arrow, ** as the **x** gets bigger the graph gets closer to the x-axis, which has a height of zero. So as **x** approaches infinity **f(x)** is approaching 0. This is called //Limit at inifinity// the Green arrow as **X** gets really small the graph gets closer to the x-axis, which is 0

Take a look as these examples!

The limit equals 3, **//SO//**

You cannot simplify, for it is in its simplistic form. So, you have to substitute.

//Does the limit exist?// //Well,// it depends. We have to look to see what happens as X approaches 3 from the left, as well as X approaches 3 from the right.

Let's take a look at the Graph of the equation, As X approaches 3 from the left (-) the graph goes down to negative Infinity. As X approaches 3 from the right (+), the graph goes up to positive Infinity. ** Since these graphs go into two different directions (two different infinities), the limit does not exist! ** __// When will the limit exist? //__ The limit will exist if the two graphs go in the same direction. Either both go up to positive infinity, or both graphs go down to negative infinity. To get more help on when a Limit exists, go to calculus Help, or [|Click here!]


 * = **Objectives ** =
 * 1) Evaluate limits using limit properties
 * 2) Evaluate limits using simplifying and rationalizing techniques.
 * 3) Evaluate limits using the Squeeze Theorem.
 * 1) Evaluate limits using the Squeeze Theorem.

__**Properties of Limits**__ The first thing you should always try is direct substitution. If you replace x with c and get a number, that is the limit. If you replace x with c and do not get a number, then try something else.

**First, set up the equation. What happens to X, As X approaches 2.**



**Second, Does Substitution work?** NO! It does not. If substitution does not work, then try something else....Can you guess what it is? **SIMPLIFY!** Here are the steps to solve this problem, if you do not understand!
 * 1) Factor the Denominator. X^2 - 4 = (x - 2) (x + 2)
 * 2) Divide both numerator and denominator by (x - 2)
 * 3) Simplify
 * 4) Plug X into the new equation.
 * 5) Get the answer.

**Evaluating limits can be hard to understand. Feel free to [|Click here] to get a video explaining this! **

=__//**Theorem (Some Basic Limits).**//__= Let b and c be real numbers and let n be a positive integer.


 * lim //b// = //b//**
 * x--->c**


 * lim //x// = //c//**
 * x--->c**


 * lim x^n = c^n**
 * x---> c**

=__//**Theorem (Properties of Limits).**//__= Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.


 * lim f(x) = L**
 * x--->c**


 * lim g(x) = K**
 * x--->c**

lim [f(x) +/- g(x)] = L +/- K

 * x--->c**

lim [f(x) g(x)] = LK

 * x--->c**

lim f(x)/g(x) = L/K K can not = 0

 * x--->c**

lim [f(x)]^//n// = L^//n//
__//**Lets Try Some Examples!**//__ //**Ex 1: find the limit (scalar multiple)**// //**Let: lim x^3 - 3 = 5**// //**x---> 2**// //**Evaulate: lim [5 (x^3 - 3)] = 5 (5) = 25**// //**x--->2**// //**(**// we could have just used direct substitution without using the scalar multiple property. However, there may be cases where a difficult limit is given in the problem statement and then you are asked to find the limit of the difficult limit multiplied by a constant.**//)//**
 * x--->c**

__//**Ex 2: find the limit (sum or difference)**//__ //**lim [x^3 + 3]**// //**x--->3**// //**lim x^3 + lim 3 = 27 + 3 = 30**// //***substitute 3 in for x to get 27 and 3***//

//**lim [x^3 + 3] = 3^3 + 3 = 30**// //**x--->3**//
 * //Now, evaluate the limit by using direct substitution, without applying the property.//**

__//**Ex 3: find the limit (product)**//__ //**Given: lim f(x) = 4 and lim g(x) = 6**// **//x--->2//** **//lim [f(x) g(x)] = 4 (6) = 24//** **//*substitute 4 in for f(x) and 6 in for g(x)*//** __**//Ex 4: find the limit (Quotient)//**__ //**Given: lim f(x) = 12 and lim g(x) = 6**// **//x--->2//** **//Evaulate lim [f(x) / g(x)]//** **//x--->2//** **//by using the quotient property//** **//lim [f(x) / g(x)] = 12 / 6 =2//** **//*substitute 12 in for f(x) and 6 in for g(x)*//** __//**Ex 5: find the limit (power)**//__ //**lim X^3**// //**x--->-2**// //**Solution: Replace x with -2**// //**lim X^3 = (-2)^3 = -8**// //**x---> -2**// [|Our Assessment! Click here!]
 * //Evaluate lim [f(x) g(x)]//**
 * //x--->2//**
 * //by using the Product property//**