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- WE'RE ASKED TO DETERMINE THE END BEHAVIOR
OR LONG RUN BEHAVIOR
FOR EACH FUNCTION.
THE END BEHAVIOR, OR LONG RUN BEHAVIOR,
DESCRIBES THE VALUE OF THE FUNCTION OR THE Y VALUE
AS X APPROACHES THE LEFT OR APPROACHES NEGATIVE INFINITY
AND APPROACHES THE RIGHT OR POSITIVE INFINITY.
THE NOTATION WE USE FOR END OR LONG RUN BEHAVIOR
IS GIVEN HERE BELOW.
AS X APPROACHES POSITIVE INFINITY,
F OF X APPROACHES SOME VALUE
AND AS X APPROACHES NEGATIVE INFINITY,
F OF X APPROACHES SOME VALUE.
SO FIRST, AS X APPROACHES POSITIVE INFINITY,
WE'D BE MOVING RIGHT ALONG THE X AXIS.
AS WE MOVE RIGHT WE APPROACH POSITIVE INFINITY.
SO WE WANT TO KNOW AS THE FUNCTION MOVES RIGHT,
WHAT IS THE FUNCTION OR Y VALUE APPROACHING?
SO IF WE MOVE RIGHT ALONG THE GRAPH,
NOTICE THE Y VALUES START TO INCREASE WITHOUT BOUND,
OR THE FUNCTION VALUES INCREASE WITHOUT BOUND
APPROACHING POSITIVE INFINITY.
SO FOR THIS FIRST HALF, AS X APPROACHES INFINITY,
F OF X ALSO APPROACHES INFINITY.
NOW WE NEED TO CONSIDER AS X APPROACHES NEGATIVE INFINITY
OR MOVES LEFT ALONG THE X AXIS IN THIS DIRECTION,
WHAT HAPPENS TO THE FUNCTION VALUES OR Y VALUES.
NOTICE AS WE MOVE LEFT ALONG THE FUNCTION
APPROACHING X = NEGATIVE INFINITY,
THE GRAPH STARTS TO GO DOWN VERY QUICKLY,
AND, THEREFORE, THE Y VALUES OR FUNCTION VALUES
ARE APPROACHING NEGATIVE INFINITY.
SO WE SAY AS X APPROACHES NEGATIVE INFINITY,
F OF X APPROACHES NEGATIVE INFINITY.
THESE TWO DESCRIBE THE END BEHAVIOR OF LONG RUN BEHAVIOR
OF THE GIVEN FUNCTION.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
WE FIRST WANT TO DETERMINE
AS X APPROACHES POSITIVE INFINITY,
F OF X OR Y APPROACHES SOME VALUE,
AND THEN DETERMINE AS X APPROACHES NEGATIVE INFINITY
F OF X APPROACHES SOME VALUE.
AGAIN, AS X APPROACHES POSITIVE INFINITY,
WE'D BE MOVING RIGHT ALONG THE X AXIS.
SO AS WE MOVE RIGHT ALONG THE FUNCTION,
WE WANT TO DETERMINE WHAT Y VALUE WE'RE APPROACHING.
WELL, AS WE MOVE RIGHT,
NOTICE HOW THE Y VALUES START TO INCREASE WITHOUT BOUND
AND, THEREFORE, F OF X OR Y IS APPROACHING POSITIVE INFINITY.
NEXT, AS X APPROACHES NEGATIVE INFINITY,
WE'RE MOVING LEFT ALONG THE X AXIS.
WE WANT TO DETERMINE WHAT VALUE F OF X OR Y
IS APPROACHING.
SO AS WE MOVE LEFT ALONG THE FUNCTION
APPROACHING X = NEGATIVE INFINITY,
NOTICE AGAIN THE V VALUES OR FUNCTION VALUES
START TO INCREASE WITHOUT BOUND,
AND THEREFORE THE FUNCTION VALUES OR Y VALUES
ARE APPROACHING POSITIVE INFINITY.
THESE TWO STATEMENTS DESCRIBE THE END BEHAVIOR
OR LONG RUN BEHAVIOR OF THE GIVEN FUNCTION.
I HOPE YOU FOUND THESE TWO EXAMPLES HELPFUL.