- WE WANT TO FIND THE LIMITS OF THE COMPOSITE FUNCTIONS
BASED UPON THE GRAPHS OF F OF X AND G OF X.
OUR FIRST LIMIT IS THE LIMIT
AS X APPROACHES -INFINITY OF G OF F OF X.
SO WE'LL START BY CONSIDERING WHAT'S HAPPENING TO F OF X,
THE INTER FUNCTION,
AS X APPROACHES -INFINITY.
SO HERE'S THE GRAPH OF F OF X.
AS X APPROACHES -INFINITY, WE'D BE MOVING LEFT ALONG THE GRAPH
AND NOTICE HOW THE FUNCTION VALUES ARE APPROACHING
THE HORIZONTAL ASYMPTOTE OF Y EQUALS ONE FROM VALUES OF Y
THAT ARE <1.
AND SINCE WE HAVE A COMPOSITE FUNCTION,
WHERE THE OUTPUTS OF F BECOME THE INPUTS INTO G
THAT MEANS THIS LIMIT IS EQUAL TO THE LIMIT
AS X APPROACHES +1
FROM THE NEGATIVE SIDE OF THE OUTER FUNCTION G OF X.
AGAIN, SINCE THE FUNCTION VALUES OF F ARE APPROACHING +1
FROM VALUES THAT ARE <1,
THESE BECOME THE INPUTS INTO THE OUTER FUNCTION G.
SO OUR LIMIT BECOMES THE LIMIT AS X APPROACHES +1
FROM THE NEGATIVE SIDE OF G OF X.
AND NOW LOOKING AT THE GRAPH OF G OF X,
WE'LL APPROACH +1 FROM THE NEGATIVE SIDE
TO DETERMINE OUR LIMIT.
AS WE APPROACH +1 FROM THE NEGATIVE SIDE,
WE'RE APPROACHING X EQUALS ONE ALONG HERE
AND WE CAN SEE WE'RE APPROACHING THE FUNCTION VALUE OF ZERO
AND THEREFORE THE LIMIT OF OUR COMPOSITE FUNCTION IS ZERO.
NOW WE'LL CONSIDER THE LIMIT AS X APPROACHES +INFINITY
OF G OF F OF X.
SO AGAIN, WE'LL FIRST CONSIDER WHAT'S HAPPENING TO F OF X
AS X APPROACHES +INFINITY.
SO AS WE MOVE TO THE RIGHT ALONG THE GRAPH OF F
NOTICE HOW WE'RE STILL APPROACHING THE FUNCTION VALUE
OF ONE,
BUT NOW IT'S GOING TO BE FROM THE POSITIVE SIDE OF ONE
OR FROM VALUES THAT ARE >1.
AND SINCE THESE FUNCTIONS VALUES BECOME THE INPUTS
INTO FUNCTION G,
THIS IS EQUAL TO THE LIMIT AS X APPROACHES +1
FROM THE POSITIVE SIDE OF G OF X.
SO NOW LOOKING AT THE GRAPH OF G OF X
AS WE APPROACH +1 FROM THE POSITIVE SIDE,
OR RIGHT SIDE, WE'RE NOW APPROACHING +1
FROM THIS DIRECTION.
AND AGAIN, NOTICE HOW WE'RE STILL APPROACHING
THE FUNCTION VALUE OF ZERO.
THEREFORE THE LIMIT OF THIS COMPOSITE FUNCTION IS ALSO ZERO.
NEXT WE HAVE THE LIMIT AS X APPROACHES ZERO
FROM THE POSITIVE SIDE, OR RIGHT SIDE, OF G OF F OF X.
SO NOW WE'RE DETERMINING WHAT'S HAPPENING TO F OF X
AS X APPROACHES ZERO FROM THE RIGHT SIDE OR POSITIVE SIDE.
HERE'S THE GRAPH OF F.
AS WE APPROACH ZERO FROM THE RIGHT SIDE,
WE'RE MOVING ALONG THE GRAPH IN THIS DIRECTION HERE
AND NOTICE HOW THE FUNCTION VALUES ARE INCREASING
WITHOUT BOUND
AND THESE FUNCTION VALUES BECOME THE INPUTS
INTO OUR OUTER FUNCTION G,
WHICH MEANS THIS IS EQUAL TO THE LIMIT
AS X APPROACHES POSITIVE INFINITY
OF THE OUTER FUNCTION G OF X.
SO NOW GOING BACK TO THE GRAPH OF G, AS X APPROACHES +INFINITY,
WE'RE MOVING RIGHT ALONG THE GRAPH
AND THE GRAPH DOES CONTINUE TO MOVE UPWARD
AND THEREFORE THIS LIMIT WILL APPROACH +INFINITY,
WHICH REMEMBER, DOES NOT EXIST.
AND NOW FOR OUR LAST LIMIT,
WE HAVE THE LIMIT AS X APPROACHES -1
FROM THE NEGATIVE SIDE, OR LEFT SIDE, OF G OF F OF X.
SO AGAIN, WE'LL FIRST CONSIDER WHAT'S HAPPENING TO FUNCTION F
AS X APPROACHES -1 FROM THE LEFT SIDE.
SO AS WE'RE APPROACHING X EQUALS -1 FROM THE NEGATIVE SIDE,
OR LEFT SIDE, WE'RE MOVING ALONG THE GRAPH IN THIS DIRECTION HERE
AND WE'RE APPROACHING THE FUNCTION VALUE ZERO
FROM VALUES THAT ARE >0
BECAUSE WE'RE ABOVE THE X AXIS,
WHICH MEANS THIS IS GOING TO BE EQUAL TO THE LIMIT
AS X APPROACHES ZERO
FROM THE POSITIVE SIDE OF THE OUTER FUNCTION G OF X.
SO NOW LOOKING AT THE GRAPH OF G OF X,
AS WE APPROACH X EQUALS ZERO FROM THE POSITIVE SIDE.
WE'RE MOVING IN THIS DIRECTION ALONG THE GRAPH
AND NOTICE HOW THE FUNCTION VALUES ARE GOING DOWN
WITHOUT BOUND
AND THEREFORE THIS LIMIT IS EQUAL TO -INFINITY,
WHICH AGAIN DOES NOT EXIST.
OKAY, THAT'S GOING TO DO IT FOR THESE FOUR EXAMPLES.
I HOPE YOU FOUND THIS HELPFUL.