- WELCOME BACK TO PART 2 OF THE FORMAL DEFINITION
OF A LIMIT.
I THOUGHT WE SHOULD TAKE A LOOK AT A SECOND PROOF OF A LIMIT.
WE WANT TO PROVE THE LIMIT AS X APPROACHES
1 OF X SQUARED + 1 = 2.
SO BEFORE WE CAN WRITE THE PROOF,
WE HAVE TO DEVELOP A RELATIONSHIP
BETWEEN EPSILON AND DELTA.
SO THE WAY WE'RE GOING TO DO THIS IS,
START WITH WHAT WE WANT, MEANING WE WANT
THE ABSOLUTE VALUE OF F OF X - L LESS THAN EPSILON.
WE'RE GOING TO REWRITE THIS SO WE CAN DEVELOP A RELATIONSHIP
BETWEEN EPSILON AND DELTA.
SO WE'LL TRANSFORM THIS ABSOLUTE VALUE
INTO THE ABSOLUTE VALUE OF X - C.
AND THEN ONCE WE HAVE THAT RELATIONSHIP,
WE CAN ACTUALLY WRITE THE PROOF.
SO WE'LL START WITH THE ABSOLUTE VALUE
OF F OF X - L IS LESS THAN EPSILON.
AND NOW WE'LL SUBSTITUTE X SQUARED + 1 FOR F OF X.
AND WE'LL SUBSTITUTE 2 IN FOR L.
NOW, LET'S CLEAR THESE PARENTHESIS
INSIDE THE ABSOLUTE VALUE.
SO WE HAVE THE ABSOLUTE VALUE OF X SQUARED,
THIS WILL BE MINUS 1, IT'S LESS THAN EPSILON.
AND NOW THIS FACTORS.
AND NOW WE CAN WRITE THIS AS A PRODUCT OF ABSOLUTE VALUES.
WE HAVE THE ABSOLUTE VALUE OF X + 1 x THE ABSOLUTE VALUE
OF X - 1 IS LESS THAN EPSILON.
NOW, FROM HERE WE'RE GOING TO MAKE AN ASSUMPTION.
NOTICE WE'RE APPROACHING X = 1, SO WE'RE GOING TO GO AHEAD
AND ASSUME THAT X IS GREATER THAN 0
AND LESS THAN 2 MEANING X IS LESS THAN 1 UNIT
FROM THE VALUE OF C.
NOW, WE COULD ASSUME ANYTHING CENTERED AROUND X = 1,
BUT THESE VALUES WORK PRETTY WELL
BECAUSE NOW WE KNOW THAT THE ABSOLUTE VALUE OF X + 1
WOULD HAVE TO BE LESS THAN 3.
AND THE ABSOLUTE VALUE OF X - 1 WOULD HAVE TO BE LESS THAN 1,
AGAIN, IF WE'RE ASSUMING THAT X IS BETWEEN 0 AND 2.
NOW, THE REASON THAT'S HELPFUL IS NOW WE CAN REWRITE,
NOW, THE REASON THIS IS HELPFUL IS WE CAN NOW REWRITE
THIS PRODUCT, THE ABSOLUTE VALUE OF X + 1
x THE ABSOLUTE VALUE OF X - 1
WOULD HAVE TO BE LESS THAN 3 x THE ABSOLUTE VALUE OF X - 1,
WHICH WE'LL LET BE LESS THAN EPSILON.
NOW, THE REASON THIS IS IMPORTANT
IS REMEMBER THE END RESULT, WE WANT TO HAVE
THE ABSOLUTE VALUE OF X - 1.
SO IF THIS IS TRUE, LOOKING AT THE RIGHT SIDE HERE,
WE KNOW THAT THE ABSOLUTE VALUE OF X - 1
MUST BE LESS THAN EPSILON DIVIDED BY 3.
SO WE'RE GOING TO LET EPSILON DIVIDED BY 3 = DELTA.
AND HERE'S THE RELATIONSHIP THAT WE NEED
IN ORDER TO WRITE OUR PROOF.
SO NOW WE'LL TAKE THIS INFORMATION
TO THE NEXT SLIDE AND WRITE THE PROOF.
NOW, I RECOPIED SOME OF THE IMPORTANT INFORMATION
UP HERE IN RED, AND NOW WE'RE GOING TO WRITE
OUR PROOF BY WORKING OUR WAY FROM LEFT TO RIGHT
USING THIS INFORMATION.
SO WE'RE GOING TO START WITH 0 IS LESS THAN THE ABSOLUTE VALUE
OF X - C LESS THAN DELTA,
STRAIGHT FROM THE FORMAL DEFINITION.
NOW WE'RE GOING TO REPLACE C WITH 1,
SO WE'LL HAVE THE ABSOLUTE VALUE OF X - 1.
AND WE'RE GOING TO REPLACE DELTA WITH EPSILON DIVIDED BY 3.
SO NOW WE'RE GOING TO MULTIPLY EVERYTHING BY 3,
SO WE'RE GOING TO HAVE 0 IS LESS THAN 3
x THE ABSOLUTE VALUE OF X - 1 LESS THAN EPSILON.
AND SINCE THE ABSOLUTE VALUE OF X + 1 IS LESS THAN 3,
IT'S TRUE THAT THE ABSOLUTE VALUE OF X + 1
x THE ABSOLUTE VALUE OF X - 1
WOULD HAVE TO BE LESS THAN EPSILON.
AND NOW WE CAN REWRITE THIS AS THE ABSOLUTE VALUE
OF X + 1 x X - 1 IS LESS THAN EPSILON.
NOW, WE'RE GOING TO GO AHEAD AND MULTIPLY THIS,
SO WE'LL HAVE 0 LESS THAN THE ABSOLUTE VALUE
OF X SQUARED - 1 LESS THAN EPSILON.
REMEMBER, OUR GOAL IS TO SHOW THAT THE ABSOLUTE VALUE
OF F OF X - L IS LESS THAN EPSILON.
AND F OF X IS X SQUARED + 1, SO WE'LL REWRITE THIS
AS THE ABSOLUTE VALUE OF X SQUARED + 1.
BUT THEN TO MAKE THIS EQUAL TO X SQUARED - 1
WE'LL HAVE TO SUBTRACT 2, WHICH IS GOOD NEWS BECAUSE L IS 2,
SO NOW WE HAVE THE MINUS L IS LESS THAN EPSILON.
SO WE HAVE OUR PROOF, WHICH WE CAN WRITE ONE MORE TIME
IN THE FINAL FORM, 0 IS LESS THAN
THE ABSOLUTE VALUE OF F OF X - L,
WHICH IS LESS THAN EPSILON.
SO ONCE WE DEVELOP THE RELATIONSHIP
BETWEEN EPSILON AND DELTA, BE ABLE TO START
WITH THE ABSOLUTE VALUE OF X - C BEING GREATER THAN 0
AND LESS THAN DELTA.
AND FROM HERE IT FOLLOWED THAT THE ABSOLUTE VALUE
OF F OF X - L WAS LESS THAN EPSILON.
SO, THIS PROOF WASN'T QUITE AS STRAIGHTFORWARD
AS THE FIRST EXAMPLE, SO I THOUGHT IT WAS WORTH SHOWING.