12.4 Limits of Infinity

Limits of Infinity are basically asking about the horizontal asymtote. In order to find the solution to a limit as it aproaches infinity, we just need to find the horizonatal asymtote.

Finding the horizontal asymtote is basically finding where the graph will go when x becomes REALLY big.

When x becomes really big, only the first term is significant. Anything after that is so minor that it won't factor into changing the final answer, so it can just be disreguarded.

When the powers of x are the same, you are adding the same amount to both the numerator of the equation and the denominator so the ration of the x value will remain the same, so the horizontal asymtote is the ratio of the x's.

When the Power of x in the numerator is greater than the power of x in the denominator, the function will continue to increase forever, not approaching any horizontal asymtote, so therefore, the asymtote doesn't exist.

When the power of x in the numerator is less than the power of x in the denominator the horizontal asymtote is zero because the bottom will continue to become bigger and bigger, creating a smaller and smaller fraction, thus approaching zero.

To find solutions to limits as x approaches infinity, go to wolframalpha

Section 12-2 Techniques for Evaluating Limits

Previously, we had 3 ways to evaluate limits: Graphing, a Numeric Approach (using tables), and Direct Substitution.

Today, we added a few more techniques to our ever increasing bag of tricks. After all, a wise man often tells us, "Variety is the spice of life."

When given a problem such as...

You MUST factor to check for holes or asymptotes

The (x-3) terms cancel. Leaving you with...

From there, you use direct substitution and plug 3 in for x. Therefore, x=-1.

Next, we moved on to another type of situation. When given a problem such as...

You again have to factor this to check for holes/asymptotes. You may be scratching your heads and saying, "How do I factor a cubic?" Well my friends, I'll remind you of something we learned all the way back in Algebra 2. SYNTHETIC DIVISION!!!

To start, you plug the number that x is approaching into your box. In this specific case, we will use 2.

If it factors, there is a hole in the graph and it has a limit. However, if it does not factor, the graph has an asymptote and therefore a limit DOES NOT EXIST.

When you use synthetic division on the original problem,

You get a remainder of -2 meaning that it has an asymptote and therefore not a limit.

Rationalizing the Numerator

In order to solve a problem like this one,

You have to multiply it by the conjugate. Usually, you would do this to rationalize the denominator, but for limit problems you rationalize either the numerator or denominator to help you solve the problem.

This leaves you with...

From there, the x's cancel out and you are left with...

You then use direct substitution and plug 0 in for x. The problem then simplifies to 1/2.

Now for your viewing pleasure, here is a song about limits. Aspects of it hold true to my life. It may for some of you as well. Enjoy!!! :)

Here is the link (the video player wasn't working)

http://www.youtube.com/watch?v=BnHuZs7ar7I

Chapter 12.1 Part Two-Introduction to Limits

Properties of Limits

1.
lim b=b
^x→a

example
lim5=5
^x→2













2.
lim x=a
^x→a

example
lim x=2
^x→2











3.
lim xⁿ=a^n
x→a

example
lim x²=4
^x→2











4.

^x→c

example

<sup>x→8

Operations with Limits

Let f(x)=L
x→a
and
g(x)=M
x→a

Scalar Multiple
lim[c × f(x)]=c × L
x→a</sup>

Sum or Difference
lim[f(x)± g(x)]=L ± M
^x→a

Product
lim[f(x) × g(x)]=L × M
^x→a

Quotient
lim[f(x) ÷ g(x)]= L ÷ M M&ne0
^x→a

Power
lim[f(x)]ⁿ=L^n
x→a

Direct Substitution

is used when:
lim f(x)=f(a)
^x→a

• doesn't work with breaks, holes or asymptotes
  
• would be "false" on a quiz because it is not always true.
  

example
lim x^2
x→4

4²
16

Continuous

• no breaks, holes or asymptotes
  
• draw without lifting pencil
  
• lim f(x)=lim f(x)=f(a)
  ^{x→a+ x→a−}
• if the limit from the left and the limit from the right are both equal to f(a) then it is continuous

not continuous










Piecewise



lim f(x)
x→1−
2(1)-3
-1

lim f(x)
x→1+
-(1)²
-1

Since the limit coming from the right and coming from the left are equal(-1), the limit is -1

if the limits from both sides were different the limit does not exist.

For your viewing pleasure....

12.1 Intro to Limits (Part 1)

Definition of a limit: If f(x) becomes arbitrarily close to a unique number L as x approaches c from either side, the limit of f(x) as x approaches c is L.

lim f(x) = L

x--> c

Ex.

lim (3x- 2)

x--> 2
Let f(x) = 3x- 2. Then construct a table that shows values of f(x) when x is close to 2.

x           f(x)

1.9       3.7

1.99     3.97

1.999   3.997

2.0       ?

2.001   4.003

2.01     4.03

2.1       4.3

From the table, you can see that the closer x gets to 2, the closer f(x) gets to 4. So you can estimate the limit to be 4. For this equation you can substitute 2 for x to obtain the limit, so:

   lim (3x- 2) = 3(2) -2 = 4

    x--> 2

x--> 1+ (the + meaning that x approaches 1 from the right; right-hand limit)
x--> 1- (the - meaning that x approaches 1 from the left; left-hand limit)

Limits That Fail to Exist:

1.) When the one-sided limits are not equal

      ex.

   

 

 

2.) Unbounded Behavior

      ex.

When x > 100 or x<-100

3.) Oscillating Behavior

      ex.

WARNING: Use the above information with caution.

Section 9.5: The Binomial Theorem

A binomial is a polynomial that has two terms. This section will teach you how to raise a binomial to a power in a relatively quick way.

The Binomial Theorem:

nCr=

For example, you have the following toppings at a pizza store:

Pepperoni

Sausage

Onion

Ham

Bacon

You are allowed to put 2 toppings on each pizza. How many different combinations of toppings do you have?

n=5

r=2

There are 77,520 different combinations of pizza

Pascal's Triangle

Pascal's Triangle starts with row 0 and gradually gets larger.

Pascal's Triangle contains many different patterns, but we are going to focus on how the numbers in each row are similar to the coefficients when raising (x+y) to different powers.

(x+y)^0=1

(x+y)^1=1x+1y

(x+y)^2=1x^2+2xy+1y^2

(x+y)^3=1x^3+3x^2y+3xy^2+1y^3

(x+y)^4=1x^4+4x^3y+6x^2y^2+4xy^3+1y^4

(x+y)^5=1x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+1y^5

The red numbers are the coefficients and make the shape and pattern of Pascal's Triangle.

Lets try an example:

1x^5+5x^4*4+10x^3*4^2+10x^2*4^3+5x*4^4+1*4^5

*Note: The binomial is raised to the 5th power. This tells you to look at row 5 of Pascal's Triangle and the numbers in this row will be the coefficients

Find the coefficient of the term:

17C10 * (2x^2)^10 * (-3i)^7

19448 * 1024x^20 * -823543i^7

The coefficient of the term is

Final Note:

=

nCr*a^r*b^(n-r)

Geometric Sequence

A geometric sequence is different from an arithmetric sequence by it is multipying instead of adding
The Common Ratio(r) is what the previous number is being multiplied by each time
example 3,6,12,24,48,96 r=3

Arithemetric Equation v. Geometric Equation

Arithmetic

a₁ =a₁
a₂ =a₁+d
a₃ =a₁+d+d
a₄ =a₁+d+d+d
.
.
.
a_n=a₁=d(n-1)

Geometric
a_a=a₁
a₂ =a₁× r
a₃ =a₁× r × r
a₄ =a₁× r × r × r
a_n=a₁× r^n-1 -equation to find nth term of a geometric sequence

Partial Sum

S_n=a₁+a₁r+a₁r²+....a₁r^n-2+a₁r^n-1
r × S_n=a₁r+a₁r²+....a₁r^n-2+a₁r^n-1+a₁rⁿ
S_n-r × S_n=a₁-a₁rⁿ S_n(1-r)=a₁(1-rⁿ) S_n=[a₁(1-rⁿ)]÷ (1-r)


example-
12r³=96
r³=8
r=2

a₃=a₁ × r^n-1
12=a₁ × r^3-1
12=a₁ × r^2
12⁄_r²=a₁
¹²⁄₄=a₁
S_n=3(1-2¹⁰)÷(1-2)
S_n=3069

Infinite Geometric Sequence

S=a₁(¹/_(1-r))
S=^a₁/_1-r

Section 9.2 Arithmetic Sequences and Partial Sums

A sequence whose consecutive terms have a common difference is called an arithmetic sequence





Definition of Arithmetic Sequence

A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence

is arithmetic if there is a number d such that

The number d is the common difference of the arithmetic sequence

The nth Term of an Arithmetic Sequence



The nth term of an arithmetic sequence has the form



a_n = dn + c




where d is the common difference between consecutive terms of the sequence and c = a_1 - d





The Sum of a Finite Arithmetic Sequence




The sum of a finite arithmetic sequence with n terms is



S_n = (n/2)(a_1 + a_n).



Example:



Find the sum of of the integers from 1 to 100



S_n = 1+2+3+4+5+6+...+99+100


= (n/2)(a_1 + a_n)


= (100/2)(1+100)


= 50(101)


=5050