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- LET'S TAKE A LOOK AT TWO MORE LIMITS ANALYTICALLY
THAT INVOLVE A PIECE-WISE DEFINED FUNCTION.
NOTICE THAT WHEN X < 1, THE GRAPH WOULD BE Y = 2X - 1
AND WHEN X >/= 1, THE GRAPH WOULD BE Y = -X - 1.
WE WANT TO DETERMINE THE LIMIT
AS X APPROACHES +1 OF OUR FUNCTION, F OF X.
SO IF WE'RE APPROACHING FROM THE LEFT OF 1,
WE'RE APPROACHING ON THE LINE Y = 2X - 1,
AND TO THE RIGHT OF 1
WE'D BE APPROACHING 1 ON THE LINE, Y = -X - 1.
SO TO DETERMINE IF THIS LIMIT EXISTS,
WE NEED TO EVALUATE BOTH LINES AT X = 1
AND SEE IF THEY RETURN THE SAME FUNCTION VALUE.
SO FOR THE FIRST LINE WE'D HAVE Y = (2 x 1) - 1, THAT'S +1,
AND THEN FOR THE SECOND LINE WE'D HAVE -1 - 1 = -2.
SO AS WE APPROACH +1 NOTICE HOW THESE 2 LINES
ARE APPROACHING TWO DIFFERENT FUNCTION VALUES
THEREFORE THIS LIMIT DOES NOT EXIST.
IT DOESN'T MEAN ALL LIMITS FOR THIS FUNCTION DON'T EXIST.
IF WE LOOK AT A LIMIT AS X APPROACHES 3 OF THE FUNCTION,
WHEN X = 3 WE'D BE ON THE LINE Y = -X - 1
AND AS WE APPROACH 3 FROM THE LEFT AS LONG AS X >/= 1,
WE'D STILL BE ON THIS LINE AND IF WE APPROACH 3 FROM THE RIGHT,
WE'D ALSO BE ON THIS LINE
WHICH MEANS WE CAN DETERMINE THE SECOND LIMIT
BY PERFORMING DIRECT SUBSTITUTION
BY REPLACING X WITH 3 ON THE LINE Y = -X - 1.
SO WE'D HAVE -3 - 1 = -4.
AND AGAIN JUST TO EMPHASIZE THIS IDEA,
LET'S LOOK AT IT GRAPHICALLY.
SO FOR THE FIRST LIMIT, AS WE APPROACH +1 FROM THE LEFT,
WE WERE APPROACHING THE FUNCTION VALUE OF 1.
WHEN WE APPROACHED 1 FROM THE RIGHT,
WE WERE APPROACHING THE FUNCTION VALUE OF -2.
AND THAT'S THE REASON WHY THE LIMIT DID NOT EXIST.
AND THEN FOR THE SECOND LIMIT,
AS WE APPROACH +3 FROM THE LEFT AND THE RIGHT,
THE LIMIT DID EXIST
AND IT WAS EQUAL TO THE FUNCTION VALUE OF F OF 3.
SO WE COULD DETERMINE THAT LIMIT
BY PERFORMING DIRECT SUBSTITUTION.