- WE'RE ASKED TO DETERMINE ANY HORIZONTAL ASYMPTOTES
OF THE GIVEN THREE RATIONAL FUNCTIONS.
FOR THESE THREE EXAMPLES, WE'LL BE USING A SHORTCUT
TO DETERMINE THE HORIZONTAL ASYMPTOTES,
AND THIS METHOD INVOLVES ANALYZING
THE DEGREE OF THE NUMERATOR AND DENOMINATOR
TO DETERMINE ANY HORIZONTAL ASYMPTOTES.
AND WHEN DETERMINING HORIZONTAL ASYMPTOTES,
WE'RE ACTUALLY DETERMINING THE END BEHAVIOR OF THE FUNCTIONS,
SO WE'LL ALSO BE RECORDING THE END BEHAVIOR DOWN HERE BELOW.
WHEN ANALYZING THE DEGREE
OF THE NUMERATOR AND DENOMINATOR OF A RATIONAL FUNCTION,
THERE ARE THREE POSSIBILITIES.
FIRST, IF THE DEGREE OF THE NUMERATOR
IS LESS THAN THE DEGREE OF THE DENOMINATOR,
THEN THE HORIZONTAL ASYMPTOTE WILL ALWAYS BE Y = 0.
IF THE DEGREE OF THE NUMERATOR
IS EQUAL TO THE DEGREE OF THE DENOMINATOR,
THEN THE HORIZONTAL ASYMPTOTE WILL BE
Y = THE RATIO OF THE LEADING COEFFICIENTS,
AND THE LAST POSSIBILITY IS THE DEGREE OF THE NUMERATOR
IS GREATER THAN THE DEGREE OF THE DENOMINATOR.
IF THIS IS THE CASE, THERE IS NO HORIZONTAL ASYMPTOTE.
NOW, LET'S TAKE A LOOK AT EACH CASE.
LOOKING AT OUR FIRST FUNCTION,
NOTICE HOW THE NUMERATOR IS DEGREE TWO,
AND THE DENOMINATOR IS DEGREE THREE.
AND SINCE THE DEGREE OF THE NUMERATOR
IS LESS THAN THE DEGREE OF THE DENOMINATOR,
THE HORIZONTAL ASYMPTOTE WILL BE Y = 0.
NOW THE REASON IT'S Y = 0 IS BECAUSE IF WE CONSIDER,
LET'S SAY, X APPROACHING POSITIVE INFINITY,
NOTICE HOW THE DENOMINATOR IS GOING TO INCREASE FASTER
BECAUSE IT'S RAISED TO THE THIRD POWER.
SO IF WE HAVE A FRACTION WHERE THE DENOMINATOR
IS INCREASING FASTER, THE FUNCTION VALUE WILL APPROACH 0.
AND THE SAME IS TRUE AS X APPROACHES NEGATIVE INFINITY.
SO IF WE HAVE A HORIZONTAL ASYMPTOTE OF Y = 0,
THEN OUR END BEHAVIOR WILL BE AS X APPROACHES INFINITY,
F(X) APPROACHES 0, AND AS X APPROACHES NEGATIVE INFINITY,
F(X) APPROACHES 0.
LET'S VERIFY THIS GRAPHICALLY.
WELL, THE HORIZONTAL LINE Y = 0 WOULD BE THE X AXIS.
WELL, IF WE'RE ON THE FUNCTION APPROACHING POSITIVE INFINITY
OR MOVING RIGHT, NOTICE HOW THE FUNCTION VALUES
ARE APPROACHING 0 TO THE RIGHT.
AND AS WE MOVE LEFT, OR AS X APPROACHES NEGATIVE INFINITY,
NOTICE HOW THESE FUNCTION VALUES
ARE APPROACHING 0 TO THE LEFT AS WELL,
SO THIS VERIFIES OUR ANSWER.
AND NOW FOR THE NEXT FUNCTION,
NOTICE HOW THE DEGREE OF THE NUMERATOR IS 3,
AND THE DEGREE OF THE DENOMINATOR IS 2.
IF THE DEGREE OF THE NUMERATOR
IS GREATER THAN THE DEGREE OF THE DENOMINATOR,
THEN THERE ARE NO HORIZONTAL ASYMPTOTES.
TO TRY TO UNDERSTAND WHY,
LET'S CONSIDER X APPROACHING POSITIVE INFINITY.
FIRST NOTICE BOTH THE NUMERATOR AND DENOMINATOR
WOULD BE POSITIVE,
BUT BECAUSE THE NUMERATOR HAS A HIGHER DEGREE,
IT'S INCREASING FASTER.
SO IF THE NUMERATOR
IS INCREASING FASTER THAN THE DENOMINATOR,
THE FUNCTION VALUE IS GOING TO APPROACH POSITIVE INFINITY.
SO WE CAN SAY AS X APPROACHES POSITIVE INFINITY,
F(X) APPROACHES POSITIVE INFINITY,
AND THEN WHEN X APPROACHES NEGATIVE INFINITY,
BECAUSE THE DEGREE IS ODD HERE, THE NUMERATOR WOULD BE NEGATIVE,
AND THE DENOMINATOR WOULD BE POSITIVE.
THEREFORE, THE FUNCTION VALUE WOULD APPROACH NEGATIVE INFINITY
AS X APPROACHES NEGATIVE INFINITY.
AND AGAIN, LET'S VERIFY THIS GRAPHICALLY.
NOTICE IF WE'RE ON THE FUNCTION MOVING TOWARD THE RIGHT,
OR AS X APPROACHES POSITIVE INFINITY,
THE FUNCTION VALUES ARE INCREASING WITHOUT BOUND,
AND THEREFORE F(X) IS APPROACHING POSITIVE INFINITY.
AND AS WE MOVE LEFT ON THE FUNCTION,
OR AS X APPROACHES NEGATIVE INFINITY,
THE FUNCTION IS MOVING DOWNWARD WITHOUT BOUND.
AND THEREFORE F(X) IS APPROACHING NEGATIVE INFINITY.
SO THIS VERIFIES OUR SECOND ANSWER.
NOW LET'S TAKE A LOOK AT THE LAST EXAMPLE.
NOTICE IN THIS CASE THE DEGREE OF THE NUMERATOR AND DENOMINATOR
ARE THE SAME.
THEY'RE BOTH DEGREE 3.
WHEN THIS IS THE CASE,
THE EQUATION OF THE HORIZONTAL ASYMPTOTE IS Y EQUALS THE RATIO
OF THE LEADING COEFFICIENTS.
THE LEADING COEFFICIENT HERE IS 2.
THE LEADING COEFFICIENT OF THE DENOMINATOR IS 1.
2 DIVIDED BY 1 IS 2,
THEREFORE THE HORIZONTAL ASYMPTOTE IS Y = 2.
TO TRY TO UNDERSTAND THIS,
NOTICE AS X APPROACHES POSITIVE INFINITY,
THE NUMERATOR AND DENOMINATOR ARE GETTING LARGER AND LARGER,
BUT BECAUSE OF THIS 2 HERE,
THE NUMERATOR IS ALWAYS GOING TO BE APPROXIMATELY TWICE
AS LARGE AS THE DENOMINATOR,
AND THEREFORE THIS RATIO IS GOING TO APPROACH
THE VALUE OF + 2.
SO IF WE HAVE A HORIZONTAL ASYMPTOTE OF 2
THE END BEHAVIOR IN BOTH CASES WILL BE F(X) APPROACHING 2.
LET'S GO AHEAD AND VERIFY THIS LAST EXAMPLE.
HERE'S THE HORIZONTAL LINE Y = 2.
NOTICE AS THEY MOVE RIGHT, OR AS X APPROACHES POSITIVE INFINITY,
THE FUNCTION VALUE APPROACHES TWO, AND AS THEY MOVE LEFT,
OR AS X APPROACHES NEGATIVE INFINITY,
THE FUNCTION VALUES ALSO APPROACH 2.
AND THAT'LL DO IT FOR THIS EXAMPLE.
I HOPE YOU FOUND THIS HELPFUL.