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- WE WANT TO DETERMINE THE ONE-SIDED LIMITS
AND ALSO DETERMINE THE EQUATIONS OF THE VERTICAL ASYMPTOTES.
BEFORE WE DO THIS THOUGH,
LET'S REVIEW 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
OR NEGATIVE SIDE OF F OF X = +/- 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 TWO LIMITS = +/- INFINITY,
THEN X = "A" IS A VERTICAL ASYMPTOTE.
SO GOING BACK TO OUR EXAMPLES,
WE FIRST HAVE THE LIMIT AS X APPROACHES 0 FROM THE LEFT,
THE NEGATIVE SIDE OF (X SQUARED - X) - 2
DIVIDED BY (X SQUARED - 3X).
LOOKING AT THE GRAPH OF THE FUNCTION HERE ON THE RIGHT,
SO IF WE'RE APPROACHING 0 FROM THE LEFT,
WE'RE APPROACHING X FROM THIS DIRECTION HERE.
AS WE GET CLOSER AND CLOSER TO 0,
NOTICE HOW THE FUNCTION VALUES DECREASE WITHOUT BOUND
APPROACHING NEGATIVE INFINITY.
AND THEREFORE OUR LIMIT IS EQUAL TO NEGATIVE INFINITY
WHICH TELLS US THE LIMIT DOES NOT EXIST,
BUT BECAUSE IT IS EQUAL TO NEGATIVE INFINITY,
THIS ALSO TELLS US THAT X = 0 IS A VERTICAL ASYMPTOTE.
ANOTHER WAY TO SHOW THIS
WOULD BE TO CREATE A TABLE OF VALUES
AND APPROACH 0 FROM THE LEFT.
SO LOOKING AT THIS FIRST TABLE, NOTICE HOW WE'RE APPROACHING 0
FROM THE NEGATIVE SIDE OR LEFT SIDE.
THIS LAST VALUE HERE IS WRITTEN IN SCIENTIFIC NOTATION,
THIS MEANS -0.0001.
BUT LOOKING AT THE FUNCTION VALUES,
NOTICE HOW THEY'RE DECREASING WITHOUT BOUND
APPROACHING NEGATIVE INFINITY, VERIFYING OUR LIMIT.
SO GOING BACK TO OUR GRAPH,
WE HAVE A VERTICAL ASYMPTOTE HERE WHERE X = 0
WHICH WOULD BE THE Y-AXIS
WHICH IS A VERTICAL LINE THE GRAPH APPROACHES,
BUT NEVER CROSSES.
NOW LOOKING AT OUR SECOND LIMIT,
WE HAVE THE LIMIT AS X APPROACHES 3 FROM THE RIGHT
OR FROM THE POSITIVE SIDE OF THE SAME FUNCTION.
SO GOING BACK OVER TO OUR GRAPH,
IF WE APPROACH +3 FROM THE RIGHT,
WE'RE APPROACHING FROM THIS DIRECTION HERE.
AS WE GET CLOSER AND CLOSER TO +3 FROM THIS DIRECTION,
NOTICE HOW THE FUNCTION VALUES ARE NOW INCREASING WITHOUT BOUND
APPROACHING POSITIVE INFINITY.
AND BECAUSE THIS LIMIT IS EQUAL TO POSITIVE INFINITY,
THIS TELLS US THE LIMIT DOES NOT EXIST,
BUT IT ALSO TELLS US
THAT WE HAVE A VERTICAL ASYMPTOTE AT X = 3.
AND AGAIN, WE CAN SHOW THIS USING A TABLE OF VALUES.
LOOKING AT THE SECOND TABLE,
NOTICE AS WE APPROACH +3 FROM VALUES THAT ARE LARGER THAN 3,
THIS DIRECTION HERE,
NOTICE HOW THE FUNCTION VALUES ARE INCREASING WITHOUT BOUND
APPROACHING POSITIVE INFINITY.
SO WE HAVE VERTICAL ASYMPTOTE HERE AT X = 3.
AND ONE LAST THING TO MENTION,
WE KNOW FROM OUR STUDY OF ALGEBRA
THAT IF WE HAVE A RATIONAL FUNCTION
AND WE FACTOR THE NUMERATOR AND DENOMINATOR,
THE VALUES OF X THAT MAKE THE DENOMINATOR 0
THAT DON'T ALSO MAKE THE NUMERATOR 0
WOULD GIVE US VERTICAL ASYMPTOTES.
NOTICE HERE, X = 0 AND X = 3
MAKE THE DENOMINATOR EQUAL TO 0 AND NOT THE NUMERATOR,
AND THEREFORE WE HAVE VERTICAL ASYMPTOTES AT X = 0 AND X = 3.
I HOPE YOU FOUND THIS HELPFUL.