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- WE WANT TO USE THE GIVEN PIECEWISE DEFINED FUNCTION
TO DETERMINE THE FOLLOWING LIMITS.
WE FIRST HAD THE LIMIT
AS X APPROACHES 1 FROM THE LEFT OF F OF X.
WELL, IF X IS APPROACHING +1
FROM THE LEFT OF THE NEGATIVE SIDE,
WE'RE APPROACHING 1 USING VALUES THAT ARE 1,
WHICH WOULD FALL INTO THIS INTERVAL HERE WHERE X IS >1.
SO WE CAN DETERMINE THIS ONE-SIDED LIMIT
BY PERFORMING DIRECT SUBSTITUTION
INTO THE FUNCTION RULE F OF X = 2X - 1.
SO WHEN SUBSTITUTING 1 FOR X WE WOULD HAVE 2 x 1 - 1.
WELL, 2 - 1 = 1.
SO THE LIMIT, AS X APPROACHES 1 FROM THE RIGHT OF F OF X = +1.
AND THEN FINALLY,
WE HAVE THE LIMIT AS X APPROACHES 1 OF F OF X.
THIS IS NOT A ONE-SIDED LIMIT.
FOR THIS LIMIT TO EXIST
THE LEFT SIDED AND RIGHT SIDED LIMIT,
AS X APPROACHES 1, MUST BE THE SAME.
AND WE CAN SEE FROM THE FIRST TWO LIMITS, THEY'RE NOT EQUAL,
AND THEREFORE THIS THIRD LIMIT DOES NOT EXIST.
NOW LET'S VERIFY THESE THREE RESULTS GRAPHICALLY.
HERE'S THE GRAPH OF OUR PIECEWISE DEFINED FUNCTION.
FOR THE FIRST LIMIT WE WE'RE APPROACHING X = +1
FROM THE LEFT SIDE OR THE NEGATIVE SIDE.
SO WE WERE APPROACHING ALONG THIS FUNCTION HERE.
AND NOTICE HOW THE FUNCTION VALUE IS APPROACHING +2.
AND THEN FOR OUR SECOND ONE-SIDED LIMIT,
WE WERE APPROACHING +1 FROM THE RIGHT SIDE,
OR FROM THE POSITIVE SIDE,
SO WE'D BE APPROACHING X = 1 ALONG THIS LINE.
NOTICE, AS WE APPROACH +1 FROM THE RIGHT,
WE'RE APPROACHING THE FUNCTION VALUE OF +1 THIS TIME.
SO THIS VERIFIES BOTH OF OUR ONE-SIDED LIMITS.
AND IT ALSO VERIFIES
THAT THE LIMIT AS X APPROACHES +1 DOES NOT EXIST
BECAUSE THE LIMITS FROM THE LEFT AND THE RIGHT
ARE NOT EQUAL TO EACH OTHER.
OKAY, I HOPE YOU FOUND THIS EXPLANATION HELPFUL.