There are two kinds of limits involving the idea of infinity… 1) Where the limit DNE or is +/- infinity 2) Where there is a limit as the value approaches +/-

infinityType 1 Example:

Note: If the problem does not specify a right or left hand limit, you must check both to make sure they equal each other.

• Our friend Patrick will show us an example of how to solve a limit problem at infinity without the use of shortcuts

• The important component for limits at infinity (with rational polynomials) is to divide by the HIGHEST POWER

• However, there are a few shortcuts… 1) Degree of Numerator=Degree of Denominator Limit (as x approaches infinity) = Ratio of Leading Coefficients of Highest Degree

2) Degree of Numerator<Degree of DenominatorLimit (as x approaches infinity)= 3) Degree of Numerator>Degree of DenominatorLimit (as x approaches infinity)=

An asymptote of a graph is a line where the function does not ever cross

A horizontal asymptote looks like this:A vertical asymptote looks like this: Horizontal and vertical asymptotes are prevalent in limit problems….For example: This graph could be used with problem that asks… what is the limit as x approaches 0??The answer?? DNEWhy??

Origin X-axis

Y-Axis

Quadrant 1: (+,+)Quadrant 2: (-,+)Quadrant 3: (-,-)Quadrant 4: (+,-)

Point P corresponds to the pair (a,b).

Fun Fact: Rene Decartes invented the Cartesian coordinates.

Y-intercept

X-intercept

-1

-2

-2

-1

Y-scale

X-scale

Coach told Johnny to run a route: 2 steps forward, cut right 4 steps, up 3 steps, and to the right again 6 steps. Make a table of the (x,y) pairs that are on the path of the football, if it is thrown to Johnny at the end of his route.

x y

0 0

2 1

4 2

6 3

8 4

10 5

y=x^2 y=x^3

y=√(x) y=IxI

Types of Graphs and Real Life Example

Vertical Compression by 1/2

Horizontal Compression by 1/2

Vertical Stretch by 2

Horizontal Stretch by 2

Reflection over x-axis

Reflection over y-axis

Shift right 1

Shift left 1

Shift up 1

Shift down 1

1/2f(x)

f(2x)

2f(x)

f([1/2]x)

-f(x)

f(-x)

f(x-1)

f(x+1)

f(x)+1

f(x)-1

f(x)

-2f(x)

f(2x)

f(-2x+8)

Real Life Example:

Timmy is mapping a routefrom his house to his school,which is across the fieldbehind his house. He drawsa line following his route,represented by the functionf(x), but he is moving twohouses down the street.What should his new functionbe to represent his route?

Red is school, blue is the oldhouse, yellow is the newhouse

Answer: f(2x)

The most important thing we learned is how to solve limits involving infinity. First, we learned the longer way and the reasoning behind the math. Then, we learned a shortcut to speed up the process.

Sources:

http://www.mathsisfun.com/definitions/quadrant-graph-.htmlhttp://accelerateu.org/resourceguides/math/m8_38.gifhttp://jwilson.coe.uga.edu/EMAT6680Fa07/Gilbert/Assignment%202/Gayle-2_files/image004.pnghttp://upload.wikimedia.org/wikibooks/en/archive/2/26/20061012185938!Y%3DX%5E3.svghttp://jmckennonmth212s09.files.wordpress.com/2009/03/sqroot-of-x1.png%3Fw%3D354%26h%3D379http://media1.shmoop.com/images/algebra-ii/alg2_ch2_narr_graphik_31.pnghttp://kartoweb.itc.nl/geometrics/Bitmaps/2D%20Cartesian%20coordinate%20system.gifhttp://mathworld.wolfram.com/images/eps-gif/Interval_1000.gifhttp://accelerateu.org/resourceguides/math/m8_38.gif

Textbookhttp://www.sosmath.com/calculus/limcon/limcon04/limcon04.htmlhttp://www.mathsisfun.com/calculus/limits-infinity.htmlhttp://tutorial.math.lamar.edu/Classes/CalcI/LimitsAtInfinityI.aspx-Patrick JMT youtube channel