- IN THIS LESSON, WE'LL TAKE A LOOK AT TWO LIMITS
AT INFINITY INVOLVING A POLYNOMIAL FUNCTION.
DETERMINING LIMITS AT INFINITY OF A POLYNOMIAL FUNCTION
IS THE SAME AS DETERMINING THE END BEHAVIOR
OR LONG RUN BEHAVIOR OF A POLYNOMIAL FUNCTION.
FIRST WE HAVE THE LIMIT AS X APPROACHES
POSITIVE INFINITY OF -4X CUBED + 5X SQUARED.
SO WHAT WE'RE TRYING TO DO HERE IS DETERMINE WHAT'S HAPPENING
TO THE FUNCTION VALUE AS X INCREASES
IN THE POSITIVE DIRECTION WITHOUT BOUND.
AND THERE ARE A COUPLE WAYS OF DOING THIS.
NOTICE HOW THE FIRST TERM HAS DEGREE 3
AND THE SECOND TERM HAS DEGREE 2.
SO AS X INCREASES WITHOUT BOUND, BECAUSE THIS FIRST TERM
HAS A HIGHER DEGREE, IT'S GOING TO OUTWEIGH
THE TERMS OF LOWER DEGREE.
THEREFORE WE CAN DETERMINE THIS LIMIT
BY JUST CONSIDERING THE TERM WITH THE HIGHEST DEGREE,
OR IN THIS CASE, THE TERM -4X CUBED.
AS X INCREASES WITHOUT BOUND, X TO THE THIRD IS GOING TO BE
A VERY, VERY LARGE POSITIVE NUMBER,
BUT THEN WE HAVE TO MULTIPLY THAT VERY LARGE POSITIVE NUMBER
TIMES -4, MAKING IT A VERY LARGE NEGATIVE NUMBER.
THEREFORE THIS LIMIT IS EQUAL TO NEGATIVE INFINITY,
WHICH WE KNOW DOES NOT EXIST.
IF THIS ARGUMENT ISN'T CONVINCING ENOUGH,
ANOTHER METHOD IS TO FACTOR OUT THE HIGHEST POWER OF X
IN THE POLYNOMIAL FUNCTION.
SO IN THIS CASE, WE WOULD FACTOR OUT X CUBED
FROM EACH TERM IN THE POLYNOMIAL.
SO LET'S GO AHEAD AND DO THAT.
WE'D HAVE THE LIMIT AS X APPROACHES POSITIVE INFINITY
AND THEN WE'RE GOING TO FACTOR OUT X TO THE THIRD.
THAT'S GOING TO LEAVE US WITH -4 PLUS,
NOW, THIS IS A LITTLE TRICKY.
IF WE FACTOR OUT 3 FACTORS OF X FROM 5X SQUARED,
THIS WILL LEAVE US WITH 5 DIVIDED BY X.
NOTICE IF WE MULTIPLY X TO THE THIRD x 5 DIVIDED BY X,
ONE FACTOR OF X WOULD SIMPLIFY OUT
LEAVING US WITH 5X SQUARED.
AND NOW BECAUSE WE HAVE A PRODUCT HERE,
WE CAN WRITE THIS AS A PRODUCT OF TWO LIMITS.
WE'D HAVE TO LIMIT AS X APPROACHES INFINITY OF X
TO THE THIRD TIMES THE LIMIT AS X APPROACHES
INFINITY OF -4 + 5 DIVIDED BY X.
AND NOW IN THIS FORM IT SHOULD BE EASIER TO SEE THAT
THE LIMIT AS X APPROACHES INFINITY
OF X CUBED WOULD APPROACH POSITIVE INFINITY,
BUT FOR THE SECOND LIMIT, NOTICE THAT -4 IS NOT AFFECTED
BY X AS IT APPROACHES INFINITY, BUT 5 DIVIDED BY X
WOULD APPROACH 0.
SO THE SECOND LIMIT WOULD BE EQUAL TO -4,
THEREFORE A VERY LARGE POSITIVE NUMBER
TIMES A NEGATIVE NUMBER DOES CONFIRM OUR LIMIT
WOULD BE NEGATIVE INFINITY.
WE'LL VERIFY THIS GRAPHICALLY AFTER WE TAKE A LOOK
AT OUR SECOND LIMIT.
NOW WE HAVE THE LIMIT AS X APPROACHES
NEGATIVE INFINITY OF THE SAME POLYNOMIAL FUNCTION,
-4X CUBED + 5X SQUARED.
AGAIN, WE ARE ABLE TO DETERMINE THIS LIMIT
JUST BY CONSIDERING THE TERM WITH THE HIGHEST DEGREE,
WHICH AGAIN IS THIS FIRST TERM.
AS X APPROACHES NEGATIVE INFINITY,
WE'RE NOW CUBING A VERY LARGE NEGATIVE NUMBER.
AND SINCE IT'S BEING RAISED TO AN ODD POWER, IT WOULD STILL
BE A VERY LARGE NEGATIVE NUMBER, BUT THEN WE'D MULTIPLY
THAT LARGE NEGATIVE NUMBER BY -4, MAKING IT AN EVEN
LARGER POSITIVE NUMBER.
AND THEREFORE THIS LIMIT IS GOING TO EQUAL
POSITIVE INFINITY, WHICH, AGAIN, DOES NOT EXIST.
AND, AGAIN, IF THIS ISN'T CONVINCING ENOUGH,
WE CAN GO AND FACTOR OUT X CUBED AS WE DID IN THE FIRST EXAMPLE.
SO LET'S GO AHEAD AND SHOW THAT AGAIN.
SO WE'LL FACTOR OUT X CUBED FROM BOTH TERMS LEAVING US
WITH -4 + 5 DIVIDED BY X.
NOW, WRITE THIS AS A PRODUCT OF TWO LIMITS.
NOW THAT WE HAVE THIS PRODUCT, IT SHOULD BE EASIER TO SEE
THAT THE LIMIT AS X APPROACHES NEGATIVE INFINITY OF X CUBED,
BECAUSE WE'RE RAISING A NEGATIVE NUMBER
TO AN ODD POWER, THIS WOULD APPROACH NEGATIVE INFINITY.
AND THE SECOND LIMIT, THE -4 IS NOT AFFECTED BY X
AND 5 DIVIDED BY X WOULD STILL APPROACH 0.
SO THE SECOND LIMIT WOULD BE EQUAL TO -4.
AND IN THIS FORM WE CAN TELL THAT
A VERY LARGE NEGATIVE NUMBER TIMES ANOTHER NEGATIVE NUMBER
WOULD BE A VERY LARGE POSITIVE NUMBER CONFIRMING OUR LIMIT
IS POSITIVE INFINITY AS X APPROACHES NEGATIVE INFINITY.
NOW, LET'S GO AHEAD AND GRAPH OUR POLYNOMIAL FUNCTION
TO VERIFY THESE LIMITS.
HERE'S THE GRAPH OF THE POLYNOMIAL FUNCTION.
NOW, AS X APPROACHES POSITIVE INFINITY,
WE'D BE MOVING TO THE RIGHT, BUT NOTICE HOW
AS THEY MOVE RIGHT THE FUNCTION GOES DOWN VERY QUICKLY.
AND THEREFORE THE FUNCTION VALUES
ARE APPROACHING NEGATIVE INFINITY,
WHICH VERIFIES OUR FIRST LIMIT.
AND THEN AS X APPROACHES NEGATIVE INFINITY,
WE'RE MOVING LEFT ALONG THE GRAPH,
AND WE CAN SEE AS WE MOVE LEFT, THE GRAPH MOVES UP VERY QUICKLY
AND GOES UP WITHOUT BOUND, APPROACHING POSITIVE INFINITY,
WHICH DOES VERIFY OUR SECOND LIMIT AS WELL.
THAT'LL DO IT FOR THIS EXAMPLE. I HOPE YOU FOUND THIS HELPFUL.