2.  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 +/- infinity Type 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.

3. 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 Denominator Limit (as x approaches infinity)= 3) Degree of Numerator>Degree of Denominator Limit (as x approaches infinity)=

4.  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?? DNE Why??

5. 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 -2 Y-scale X- scale

6. 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. xy 00 21 42 63 84 105 y=x^2 y=x^3 y=√(x) y=IxI Types of Graphs and Real Life Example

7.  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

8. f(x) -2f(x) f(2x) f(-2x+8)

9.  Real Life Example: Timmy is mapping a route from his house to his school, which is across the field behind his house. He draws a line following his route, represented by the function f(x), but he is moving two houses down the street. What should his new function be to represent his route? Red is school, blue is the old house, yellow is the new house Answer: f(2x)

10.  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-.html http://accelerateu.org/resourceguides/math/m8_ 38.gif http://jwilson.coe.uga.edu/EMAT6680Fa07/Gilbert/ Assignment%202/Gayle-2_files/image004.png http://upload.wikimedia.org/wikibooks/en/archive /2/26/20061012185938!Y%3DX%5E3.svg http://jmckennonmth212s09.files.wordpress.com/2 009/03/sqroot-of-x1.png%3Fw%3D354%26h%3D379 http://media1.shmoop.com/images/algebra- ii/alg2_ch2_narr_graphik_31.png http://kartoweb.itc.nl/geometrics/Bitmaps/2D%20 Cartesian%20coordinate%20system.gif http://mathworld.wolfram.com/images/eps- gif/Interval_1000.gif http://accelerateu.org/resourceguides/math/m8_ 38.gif Textbook http://www.sosmath.com/calculus/li mcon/limcon04/limcon04.html http://www.mathsisfun.com/calculus /limits-infinity.html http://tutorial.math.lamar.edu/Classe s/CalcI/LimitsAtInfinityI.aspx -Patrick JMT youtube channel