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Direct substitution

example:

Dividing out

example:

direct substitution would not work here because when you but -3 in for x the denominator will equal zero.

to use dividing out you would first factor

(x+3) will cancel out

now use Direct Substitution

Rationalizing

example:

if you use direct substitution...

first you have to rationalize the sqrt

X cancels out

now use direct substitution

12.1 Limits- Cont.

Continuous Functions

• No holes, breaks, or asymptotes in the graph of the function
• A function f is continuous at point c if:
• The limit exists
• 

Example:

NOT Continuous

The right and left side limits of the function are not the same, so the limit does not exist and the function is NOT continuous.

12.1 Limits

Definition: 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

Examples:

Find the limit

1.

Limit is found by plugging in a 2 for x

2.

means as C approaches 2 from the right

means as C approaches 2 from the left

When limits do not exist (DNE)

1.

2.

3.

9.5 Binomial Theorem

(a+b)^0 = 1

(a+b)^1 = a+b

(a+b)^2 = a^2 +2ab+b^2

etc.

Binomials raised to a power that is a natural number follow the pattern of pascals triangle.

*The coefficients of the terms of the binomial (a+b)^n are the terms of pascals triangle in row "n" as they appear from left to right.

*The power that "a" is raised to starts at "n" and decreases by 1 each term until it reaches 0

*The power that "b" is raised to starts at 0 and increases by 1 until it reaches "n"

example: (a+b)^2

the terms of row 2 of pascals triangle are 1, 2, 1

(a+b)^2 = (1)a^2 + (2)ab + (1)b^2

= a^2 + 2ab + b^2

9.3 Geometric Sequences

Geometric Sequence: a sequence of numbers made by multiplying by some value each time.
- "Some value" is the common ratio (r)

ex. 3, 6, 12, 24, 48, 96, .....

The difference between each term is found by dividing by 2 each time, making the common ratio 2. 3 x 2 = 6
6 x 2 = 12, ect.

Explicit Formula
*always exponential*












so,



Ex.
3, 6, 12, 24, 48, 96, ...

r=2

so the explicit formula would be:




Finding the nth Term




To find the nth term, take the first value of the geometric sequence and multiply it by the quantity of one minus the common ratio to the nth term, divided by one minus the common ratio.

Ex.

3, 6, 12, 24, 48, ...

r=2

Finding the 6th term:






Summation Notation

Finding the nth partial sum of a geometric sequence:






Where



is the explicit formula.

9.2 Arithmetic Sequence

Arithmetic Sequence: A sequence whose consecutive terms have a common difference

d is the Common Difference of the arithmetic sequence.

The term of an Arithmetic Sequence:

The Sum of a Finite Arithmetic Sequence:

The Partial Sum of an Arithmetic Sequence:

Find the 150th partial sum of the arithmetic sequence: 5, 16, 27, 38, 49, ...

and so, the nth term is

therefore,

= 75(1649)

=123,675