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- WE WANT TO DETERMINE THE FOLLOWING TWO ONE-SIDED LIMITS.
BEFORE WE DO THIS THOUGH,
I WANT TO MAKE THE CONNECTION BETWEEN ONE-SIDED LIMITS
AND VERTICAL ASYMPTOTES.
THE VERTICAL LINE X = "A"
IS A VERTICAL ASYMPTOTE OF THE GRAPH OF F OF X
IF THE LIMIT AS X APPROACHES "A" FROM THE LEFT OF F OF X
EQUALS +/- INFINITY
OR IF THE LIMIT AS X APPROACHES "A" FROM THE RIGHT OF F OF X
IS EQUAL TO +/- INFINITY.
SO IF EITHER OF THESE LIMITS ARE EQUAL TO +/- INFINITY,
THEN X = "A" WILL BE A VERTICAL ASYMPTOTE.
SO GOING BACK TO OUR TWO EXAMPLES,
WE HAVE THE LIMIT AS X APPROACHES 2/3 FROM THE RIGHT
OR FROM THE POSITIVE SIDE OF 5X DIVIDED BY (2 - 3X).
NOTICE IN THIS CASE, DIRECT SUBSTITUTION WILL NOT WORK
BECAUSE IF WE SUBSTITUTE 2/3 FOR X IN THE DENOMINATOR,
OUR DENOMINATOR WOULD BE 0.
SO THERE ARE SEVERAL WAYS TO THINK ABOUT THIS LIMIT.
LET'S FIRST LOOK AT THE GRAPH OF THE FUNCTION.
NOTICE ON THE X-AXIS,
HERE'S WHERE X WOULD BE EQUAL TO 2/3.
SO IF WE APPROACH 2/3 FROM THE POSITIVE SIDE,
OR THE RIGHT SIDE,
WE'D BE APPROACHING 2/3 FROM THIS DIRECTION HERE.
AS WE GET CLOSER AND CLOSER TO 2/3 FROM THIS DIRECTION,
NOTICE HOW THE FUNCTION VALUES
DECREASE WITHOUT BOUND APPROACHING NEGATIVE INFINITY.
SO OUR LIMIT IS EQUAL TO NEGATIVE INFINITY
WHICH TELLS US TWO THINGS, FIRST THE LIMIT DOES NOT EXIST.
BECAUSE THE LIMIT IS EQUAL TO NEGATIVE INFINITY,
IT ALSO TELLS US X = 2/3 IS A VERTICAL ASYMPTOTE.
GOING BACK OVER TO THE GRAPH,
HERE'S THE VERTICAL LINE X = 2/3,
WHERE A VERTICAL ASYMPTOTE IS A LINE
THE GRAPH APPROACHES BUT NEVER CROSSES.
ANOTHER WAY TO DETERMINE THIS LIMIT
WOULD BE TO MAKE A TABLE OF VALUES APPROACHING 2/3
FROM THE POSITIVE SIDE.
SO FIRST 2/3 AS A DECIMAL WOULD BE 0.6 REPEATING.
SO IF WE APPROACH 2/3 FROM THE POSITIVE SIDE OR THE RIGHT SIDE,
WE'D BE APPROACHING FROM VALUES
THAT ARE GREATER THAN 0.6 REPEATING.
LOOKING AT THE TABLE OF VALUES,
NOTICE AS X APPROACHES 2/3 FROM THE POSITIVE SIDE
OR FROM VALUES LARGER THAN 2/3,
NOTICE HOW THE FUNCTION VALUES ARE DECREASING WITHOUT BOUND
APPROACHING NEGATIVE INFINITY.
SO THIS TABLE DOES VERIFY OUR LIMIT.
FOR THE NEXT LIMIT,
WE HAVE THE LIMIT AS X APPROACHES 2/3 FROM THE LEFT
OR FROM THE NEGATIVE SIDE OF THE SAME FUNCTION.
SO NOW GOING BACK OVER TO THE GRAPH,
WE'RE APPROACHING 2/3 FROM THIS DIRECTION HERE.
AS WE GET CLOSER AND CLOSER TO 2/3 FROM THIS DIRECTION,
NOTICE HOW IN THIS CASE
THE FUNCTION VALUES INCREASE WITHOUT BOUND,
NOW APPROACHING POSITIVE INFINITY.
SO BECAUSE THIS IS APPROACHING POSITIVE INFINITY,
THIS TELLS US THE LIMIT DOES NOT EXIST
AND ALSO THAT X = 2/3 IS A VERTICAL ASYMPTOTE
WHICH WE ALREADY SAW FROM THE FIRST ONE-SIDED LIMIT
WE'RE USING A TABLE OF VALUES WE WOULD NOW APPROACH 2/3
FROM VALUES THAT ARE SMALLER THAN 2/3.
OR FROM THE LEFT--NOTICE AS WE APPROACH 2/3 FROM THE LEFT,
THE FUNCTION VALUES ARE INCREASING WITHOUT BOUND
APPROACHING POSITIVE INFINITY.
THERE IS ONE MORE THING TO MENTION THOUGH,
WE SHOULD REMEMBER FROM OUR STUDY
OF RATIONAL FUNCTIONS IN ALGEBRA
THAT IF WE FIND THE VALUES OF X
THAT MAKE THE DENOMINATOR EQUAL TO 0 AND NOT THE NUMERATOR,
THIS WOULD ALSO GIVE US THE EQUATION
OF THE VERTICAL ASYMPTOTE.
SO NOTICE IN THIS CASE
IF WE WERE TO SET 2 - 3X = 0 AND SOLVE FOR X,
WE WOULD SUBTRACT 2 ON BOTH SIDES
AND DIVIDE BOTH SIDES BY -3
GIVING US THE EQUATION OF VERTICAL ASYMPTOTE X = 2/3.
I HOPE YOU FOUND THIS HELPFUL.