Tip:
Highlight text to annotate it
X
- THE LAST TYPE OF LIMIT WE WANT TO LOOK AT GRAPHICALLY
ARE LIMITS AT INFINITY.
HERE WE HAVE THE LIMIT AS X APPROACHES
POSITIVE INFINITY OF F OF X.
SO NOW INSTEAD OF X APPROACHING A SPECIFIC VALUE, OR A CONSTANT,
IT'S INCREASING WITHOUT BOUND OR APPROACHING POSITIVE INFINITY.
SO WE WANT TO DETERMINE, AS X APPROACHES
POSITIVE INFINITY ALONG THE FUNCTION,
IS THE FUNCTION APPROACHING A SPECIFIC VALUE,
WHICH MEANS IS THE FUNCTION APPROACHING A SPECIFIC Y VALUE.
SO IF THE LIMIT AS X APPROACHES POSITIVE INFINITY,
WE'RE ON THE FUNCTION APPROACHING POSITIVE INFINITY
ALONG HERE.
SO AS WE CONTINUE TO MOVE TO THE RIGHT,
WE WANT TO DETERMINE IF WE'RE APPROACHING
A SPECIFIC Y VALUE OR FUNCTION VALUE.
ONE THING I NOTICED ABOUT THIS GRAPH IS THAT
IT DOES HAVE A HORIZONTAL ASYMPTOTE
AT Y EQUALS 2.
SO AS WE CONTINUE TO MOVE TO THE RIGHT ALONG
THE FUNCTION AND X APPROACHES POSITIVE INFINITY,
THE FUNCTION WILL APPROACH THE VALUE OF 2.
THEREFORE THIS LIMIT IS EQUAL TO 2.
FOR THE SECOND LIMIT WE HAD THE LIMIT AS X APPROACHES
NEGATIVE INFINITY OF F OF X, WHICH MEANS WE'RE NOW
ON THE FUNCTION MOVING TO THE LEFT
AS X APPROACHES NEGATIVE INFINITY
IN THIS DIRECTION HERE.
AND, AGAIN, AS WE MOVE TO THE LEFT,
NOTICE HOW THE FUNCTION IS APPROACHING
THE HORIZONTAL ASYMPTOTE AND THEREFORE THE FUNCTION VALUE
IS APPROACHING 2 IN THIS DIRECTION AS WELL.
SO THE LIMIT AS X APPROACHES NEGATIVE INFINITY
OF F OF X IS EQUAL TO 2.
AND I ALSO INCLUDED A REGULAR LIMIT HERE.
WE WANT THE LIMIT AS X APPROACHES -3 OF F OF X.
AND FOR THIS LIMIT, WE'LL SKETCH A VERTICAL LINE
AT X EQUALS -3 AND WE'LL APPROACH X EQUALS -3
FROM BOTH SIDES TO SEE IF WE APPROACH
THE SAME FUNCTION VALUE.
AND, AGAIN, NOTICE HOW THERE IS A HOLE AT -3,
BUT THIS HOLE IS NOT GOING TO AFFECT WHETHER THIS LIMIT
EXISTS OR NOT.
SO AS WE APPROACH FROM THE LEFT, WE'RE APPROACHING ALONG HERE.
AS WE GET CLOSER AND CLOSER TO X EQUALS -3,
WE'D BE APPROACHING THIS POINT HERE
AND THEREFORE THE FUNCTION IS APPROACHING 1.
AND NOTICE HOW WHEN WE APPROACH FROM THE LEFT ALONG HERE,
WE'RE APPROACHING THAT SAME POINT
SO THE FUNCTION VALUE IS ALSO APPROACHING 1.
SO, SINCE WE'RE APPROACHING THE SAME FUNCTION VALUE
FROM THE LEFT HERE IN BLUE, AND FROM THE RIGHT HERE
IN GREEN, THE LIMIT AS X APPROACHES -3 OF F OF X
IS EQUAL TO 1.
LET'S TAKE A LOOK AT THREE MORE EXAMPLES.
FIRST, WE HAVE THE LIMIT AS X APPROACHES
POSITIVE INFINITY OF F OF X.
SO WE'RE ON THE FUNCTION AND MOVING TO THE RIGHT
AS X APPROACHES POSITIVE INFINITY.
SO WE'RE MOVING IN THIS DIRECTION HERE.
AND NOTICE HOW THERE IS NOT A HORIZONTAL ASYMPTOTE.
IT LOOKS LIKE THERE'S ACTUALLY A SLANT ASYMPTOTE,
BUT THE IDEA HERE IS THAT THE FUNCTION
WILL INCREASE WITHOUT BOUND AND THEREFORE
AS X APPROACHES INFINITY, THE FUNCTION WILL ALSO
APPROACH POSITIVE INFINITY.
AND THEREFORE THE LIMIT DOES NOT EXIST.
SO, AGAIN, WE'RE APPROACHING POSITIVE INFINITY.
AND THEN FOR THE SECOND EXAMPLE, WE HAVE THE LIMIT
AS X APPROACHES NEGATIVE INFINITY OF F OF X.
SO NOW WE'RE MOVING TO THE LEFT ALONG THE FUNCTION.
SO IF WE MOVE TO THE LEFT IN THIS DIRECTION HERE,
NOTICE HOW THE FUNCTION IS DECREASING WITHOUT BOUND
AND THEREFORE THE FUNCTION IS APPROACHING NEGATIVE INFINITY
AS X APPROACHES NEGATIVE INFINITY.
AND, AGAIN, THIS DOES NOT EXIST.
AND THEN FOR THIS LAST ONE, WE HAVE THE LIMIT
AS X APPROACHES -3 OF F OF X.
SO WE'LL BE APPROACHING -3 FROM THE LEFT AND THE RIGHT
TO SEE IF WE'RE APPROACHING THE SAME FUNCTION VALUE.
SO HERE'S X EQUALS -3.
SO AS WE APPROACH FROM THE LEFT, NOTICE HOW WE'RE APPROACHING
THIS POINT HERE.
SO WE'RE APPROACHING THE FUNCTION VALUE OF 0.
AND THE SAME IS TRUE WHEN WE APPROACH -3 FROM THE RIGHT.
WE'RE APPROACHING A FUNCTION VALUE OF 0,
THEREFORE THE LIMIT AS X APPROACHES 3 OF F OF X
IS EQUAL TO 0.
AND I THINK WE'LL GO AHEAD AND STOP HERE.