12.4 Limits at Infinity

Definition of Limits at Infinity:
If f is a function and the limits are real numbers, then
  and
 
These limits are read as "the limit of f(x) as x approaches is "
and "the limit of f(x) as x approaches is ."


You can use the properties of limits to evaluate a limit at infinity.
Example:





So, the limit of as x approaches is 4.


When f(x) is a rational function, and the largest term in the numerator and denominator have the same exponential value for the variable, the limit can be reduced to the coefficients of those terms.
Example:

This complicated limit can be reduced to

because you can ignore everything but the largest terms in the numerator and denominator and reduce the fraction that's left.

Evaluating Limits

Limits of Polynomial and Rational Fuctions

1.       If p is a polynomial function and c is a real number, then

     

     2. If r is a rational function given by                                                                and c is a real number, then





To find the limit of a polynomial or rational function, you can do a few things.

The first is direct substitution. This means that you take the number that x is approaching and substitute it in for x in F(x).

Example:

7(-3) +12

-9

The second is by dividing out. This means that you have to factor the numerator and denominator and divide out the common factors, and then use direct substitution.

Example:

(x+1)

1+1

2

The third is the rationalizing technique. This is when you first rationalize the numerator of a function.

Example:

Using direct substitution, this would be indeterminate, or
  However, you can rewrite the fraction by rationalizing the numerator.

 Now the problem can be solved by direct substitution. when 0 is put into the equation for x, the limit can be determined to be negative one half.

-Rachel

Arithmetic Sequences

An arithmetic sequence is a type of sequence where the numbers increase (or decrease) by a common difference d.
To find a specific term in an arithmetic sequence, we could do two things. If we had a lot of time on our hands, we could just keep adding the common difference, but that's kind of frowned on by math teachers. The easier way to find a specific term is by using this formula:

In this formula, a(n) is the actual number you're trying to find. a(1) is the first term you're given. d is the common difference, and n is the number of the term you're on.
This might be a little tricky to understand with just words, so here's an example.

Sample sequence: 4, 7, 10, 13, 16...
We're going to find the 50th term in the sequence.
The difference d between each of these terms is a positive 3 added on to each. Since we're starting our sequence at 4, a(1)=4. The term number n we're trying to find is 50. Let's plug those in to our formula:

If we solve this, we find that a(50)=151.

Now let's say that we don't just want to find out what a particular term is - we want to sum up all the numbers up to that term. This is called a partial sum, and again we have two ways to find the value. We could manually add up all the numbers (again, frowned on by math teachers), or we could find it with this formula:

 Most of the terms in this formula represent the same things they represented in the formula for a particular term, but there are a couple of small changes. First of all, S(n) is the sum you're trying to find. Second, a(n) is not the term number you're trying to find but the term number at which you stop adding.

Summation Notation

9.1: Sequences and Series- Summation Notation

Sequence:
3,7,11,15,....

Series: A series is the sum of a sequence.
3+7+11+15...

                                          (Upper Limit)

                                          (Lower Limit)

This greek letter is known as Sigma.  It represents the sum of something.  In this case it represents the sum of a series.  Sigma with an upper limit, lower limit, and explicit formula is called summation notation.


Example:

                                                            10

                                                            n=1

This is the partial sum starting with the lower limit of n=1, ending with the last term being 10 (upper limit), and using the explicit formula. 

The sum of this series = 160 when you add all of the terms of the partial sum together. 

To solve this on a calculator:

sum(seq(explicit formula (2n+5), variable (x), lower limit (1), upper limit (10))

This is a faster way to find the sum instead of adding each term up individualy. 

9.1- Sequences and Series

Domain= only NATURAL numbers


Example 1:
                   n =  1    2     3       4      5      6
                          3, 6, 11, 15, 19, 23...       
ADD 4 for next term

Recursive formula:  
                                            
                                       OR    
                                              NEXT=NOW+4

Recursive= repeated thing based on what came first 


Explicit Formula:
                                      
                                        4n= Slope or Rate of Change


Example 2:
                    n =  1   2     3       4      5      6
                         3, 6, 12, 24, 48, 96...
MULTIPLY by 2 for next term

Recursive formula:
                                        
                             
                                         NEXT=NOW* 2

Explicit Formula:
                                  

3.3: Properties of Logarithms

Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use change-of-base formulas.

Change-of-Base Formulas

Base b

Base 10

Base e

Properties of Logarithms

Let a be a positive number such that a  1, and let n be a real number. If u and v  are positive real numbers, the following properties are true.

1.                        4.

2.                            5. 

3.                                       6. 

Solving Exponential and Logarithmic Equations

Two basic strategies:

-one-to-one properties

if and only if  x=y

if and only if x=y

-inverse properties

= x

Strategies:

1- Rewrite the given equation in a form to use the one-to-one properties of exponential or logarithmic functions.

2- Rewrite an exponential equation in logarithmic form and apply the inverse property of logarithmic functions.

3- Rewrite a logarithmic equation in exponential form and apply the inverse property of exponential functions.

Example-

=5

=

=

2x=

x=

x= 0.112



Continuous Compounding-

Change of Base Formula-