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- WELCOME TO A VIDEO ON DETERMINING LIMITS
THAT INVOLVE TRIGONOMETRIC FUNCTIONS.
AND THE GOAL OF THIS VIDEO IS TO DETERMINE LIMITS
USING A VARIETY OF TECHNIQUES.
NOW IN MY VIDEO ENTITLED "INTRODUCTION TO LIMITS"
WE DO DISCUSS WHAT A LIMIT IS
AND YOU MAY WANT TO WATCH THAT VIDEO FIRST
IF YOU HAVEN'T ALREADY.
IN GENERAL, THERE ARE THREE WAYS TO APPROACH FINDING LIMITS.
THE NUMERICAL APPROACH IN WHICH WE USE A T-TABLE
TO LOOK AT VALUES OF THE FUNCTION AS X APPROACHES C,
THE GRAPHICAL APPROACH WHERE WE ANALYZE THE GRAPH
AND THEN LASTLY, THE ANALYTICAL APPROACH WHERE WE USE ALGEBRA
OR CALCULUS TECHNIQUES TO FIND LIMITS.
LET'S ALSO REVIEW THE INFORMAL DEFINITION OF A LIMIT.
THE LIMIT AS X APPROACHES C OF F OF X IS EQUAL TO L
IF ALL OF THE VALUES OF F OF X ARE CLOSE TO L FOR VALUES OF X
THAT ARE SUFFICIENTLY CLOSE, BUT NOT EQUAL TO C.
AND SUFFICIENTLY CLOSE WOULD CONSIST OF VALUES
THAT ARE LESS THAN OR GREATER THAN C.
AND OFTEN WE SAY VALUES THAT ARE TO THE LEFT
AND TO THE RIGHT OF C.
SOME FUNCTIONS MAY NOT HAVE LIMITS AT SPECIFIC VALUES OF C,
WHICH WE'LL SEE IN JUST A SECOND.
AND LASTLY, NOTICE THE VALUE OF THE LIMIT IS NOT AFFECTED
BY THE VALUE OF THE FUNCTION AT X EQUALS C.
NOW, THERE ARE THREE TYPES OF BEHAVIOR
FOR THE NON-EXISTENCE OF A LIMIT.
THE FIRST IS THAT THE FUNCTION APPROACHES A DIFFERENT NUMBER
OR A DIFFERENT VALUE FROM THE LEFT AND RIGHT SIDE OF C.
NUMBER TWO, F OF X INCREASES OR DECREASES WITHOUT BOUND
AS X APPROACHES C.
AND LASTLY, F OF X OSCILLATES BETWEEN FIXED VALUES
AS X APPROACHES C.
LET'S GO AHEAD AND LOOK AT A FEW EXAMPLES.
THE LIMIT AS X APPROACHES PI/6 OF SIGN X.
WELL, HERE'S THE GRAPH OF Y EQUALS SIGN X.
LETS GO AHEAD AND IDENTIFY WHERE X IS EQUAL TO PI/6.
AND THAT WOULD BE THIS VERTICAL LIN HERE.
SO WE WANT TO SEE
IF WE'RE APPROACHING THE SAME FUNCTION VALUE
FROM THE RIGHT AND FROM THE LEFT.
WELL, SINCE THE SIGN FUNCTION IS NICE AND SMOOTH
AND CONTINUOUS AROUND PI/6
THAT TELLS US THAT WE CAN PERFORM DIRECT SUBSTITUTION
TO FIND THE VALUE OF THIS LIMIT.
SO THIS LIMIT IS EQUAL TO THE SIGN OF PI/6
AND THIS SHOULD LOOK VERY FAMILIAR TO US.
PI/6 IS 30 DEGREES,
SO A 30-60-90 RIGHT TRIANGLE WOULD LOOK SOMETHING LIKE THIS
WHERE THE SIGN OF 30
IS EQUAL TO THE OPPOSITE OVER THE HYPOTENUSE,
WHICH IS EQUAL TO 1/2.
SO THIS LIMIT IS EQUAL TO 1/2.
AND THESE ARE THE KIND OF LIMITS THAT WE LIKE
BECAUSE WE CAN FIND THEM BY PERFORMING DIRECT SUBSTITUTION,
BUT IN MANY CASES WE CANNOT DO THIS.
THE LIMIT AS X APPROACHES PI/2 OF TANGENT X.
AGAIN, LETS IDENTIFY GRAPHICALLY WHERE X IS EQUAL TO PI/2.
THAT WOULD BE THIS VERTICAL LINE HERE.
SO WE'RE APPROACHING THIS VERTICAL LINE
FROM THE RIGHT SIDE IN THIS DIRECTION
AND THE LEFT SIDE IN THIS DIRECTION.
SO RIGHT AWAY, WE CAN SEE
THAT WE'RE NOT APPROACHING THE SAME VALUE.
AND IN FACT THE LIMIT AS X APPROACHES PI/2 FROM THE RIGHT
WOULD BE EQUAL TO NEGATIVE INFINITY.
AND THE LIMIT AS X APPROACHES PI/2 FROM THE LEFT
WOULD BE EQUAL TO POSITIVE INFINITY.
SO OBVIOUSLY, THESE DO NOT APPROACH THE SAME VALUE
AND THEY ALSO INCREASE AND DECREASE WITHOUT BOUND,
THEREFORE THIS LIMIT DOES NOT EXIST.
LET'S GO AHEAD AND TAKE A LOOK AT ONE MORE.
THE LIMIT AS X APPROACHES ZERO OF SIGN 1/X.
GRAPHICALLY, AS X APPROACHES ZERO
WE CAN SEE SOMETHING ODD IS HAPPENING IN THIS REGION.
LET'S TAKE A CLOSER LOOK AT THIS BY USING A T-TABLE.
WHAT I WANT TO DO HERE IS TAKE A LOOK AT THE VALUE OF 1/X,
WHICH IS THE ARGUMENT WE'RE TAKING THE SIGN OF.
SO THIS IS NOT THE FUNCTION VALUE,
THIS IS JUST THE ARGUMENT OF 1/X.
SO IF X IS EQUAL TO -0.1, THIS IS THE SAME AS -1/10.
1 DIVIDED BY -1/10 IS THE SAME
AS 1 x -10/1 THIS WOULD GIVE A VALUE OF -10.
AND FOR THE SAME REASON, WHEN X IS +0.1,
THIS WOULD BE EQUAL TO 10.
NEXT, WHEN X IS EQUAL TO -0.01 OR -1/100.
1/X WOULD BE EQUAL TO -100 AND THIS WOULD BE +100.
AND THESE LAST TWO COLUMNS, AS X GETS EVEN CLOSER TO ZERO
AND TAKES ON THE VALUE OF - 0.001, WHICH IS -1/1000.
1/X WOULD BE EQUAL TO -1000 AND THIS WOULD BE EQUAL TO A 1000.
THE NEXT THING I WANT YOU TO THINK ABOUT
IS THAT THE PERIOD FOR THIS SIGN FUNCTION IS EQUAL TO 2PI
OR ROUGHLY 6.28.
SO EVERY 6.28 RADIANS THE SIGN FUNCTION
TAKES ON ALL THE VALUES BETWEEN -1 AND +1.
SO GOING BACK TO OUR TABLE, AS X APPROACHES ZERO,
1/X IS ACTUALLY APPROACHING POSITIVE AND NEGATIVE INFINITY,
WHICH MEANS THAT REMEMBER EVERY 6.2 RADIANS,
THE FUNCTION VALUE WILL ALTERNATE BETWEEN -1 AND +1.
SO AS WE GET CLOSER AND CLOSER TO ZERO,
THE FUNCTION SIGN OF 1/X IS ALTERNATING BETWEEN -1 AND +1
FASTER AND FASTER AND FASTER.
AND THEREFORE, BECAUSE THIS FUNCTION ALTERNATES
BETWEEN -1 AND +1
AS X APPROACHES ZERO, THIS LIMIT DOES NOT EXIST.
LET'S TAKE A LOOK FOR THIS ON OUR GRAPHING CALCULATOR AS WELL.
SO IN Y1 WE HAVE SIGN 1/X, LET'S ADJUST OUR WINDOW
SO THAT THE X INTERVAL IS BETWEEN -3 AND +3.
AND THE Y INTERVAL IS BETWEEN -2 AND +2.
LET'S GO AHEAD AND GRAPH IT.
AND AGAIN, WE CAN SEE AS X GETS CLOSER TO ZERO,
IT STARTS TO ALTERNATE.
LET'S GO AHEAD AND ZOOM IN ON THIS
AND SEE WHAT WE CAN DISCOVER.
SO IF WE PRESS ZOOM, OPTION TWO, AND THEN ENTER.
AGAIN YOU CAN SEE AS WE'RE APPROACHING ZERO,
THIS FUNCTION IS ALTERNATING BETWEEN -1 AND +1.
LET'S GO AHEAD AND ZOOM IN AGAIN.
AND OF COURSE, WE CAN KEEP DOING THIS
OVER AND OVER AND OVER AGAIN,
BUT IT'S BECOMING VERY CLEAR,
I HOPE, THAT THIS FUNCTION IS ALTERNATING BETWEEN -1 AND +1.
LET'S GO AHEAD AND TRY IT ONE MORE TIME.
ZOOM IN, ENTER
AND IT'S JUST ALTERNATING FASTER AND FASTER BETWEEN -1 AND +1.
OKAY, I HOPE YOU FOUND THESE EXPLANATIONS HELPFUL.
THANK YOU AND HAVE A GOOD DAY.