- HELLO, AND WELCOME 
TO AN INTRODUCTION TO LIMITS.
THE GOALS OF THIS VIDEO 
ARE TO GIVE
AN INFORMAL DEFINITION 
OF A LIMIT,
AND ALSO TO FIND LIMITS, 
AND ONE-SIDED LIMITS
GRAPHICALLY AND NUMERICALLY.
SOMETIMES IT'S DIFFICULT 
TO PUT YOUR FINGER
ON THE EXACT MEANING OF A LIMIT, 
SO WHAT I'D LIKE TO DO
IS START WITH AN ANALOGY.
LET'S SAY, FOR EXAMPLE, 
YOU'RE AT POSITION ONE,
AND THERE'S A FAN 
AT POSITION FOUR.
IF YOU WANT TO KNOW 
THE SPEED OF THE WIND
PRODUCED BY THE FAN 
AS X GETS CLOSE TO FOUR,
THE EXACT POSITION OF THE FAN, 
WE WOULD MEASURE THE WIND SPEED
AS WE GET CLOSER AND CLOSER 
TO X = 4.
THE PROBLEM IS WE CANNOT 
GET TO THE EXACT POSITION X = 4
OR WE WOULD RISK 
SERIOUS BODILY INJURY.
AS A RESULT, WE COULD USE 
THE DATA LISTED ABOVE
TO CONCLUDE THAT THE WIND SPEED 
OF THE FAN AT X = 4
IS APPROACHING 
SIX MILES PER HOUR.
LET'S TAKE A LOOK AT OUR TABLE.
IF WE START WITH AN X VALUE OF 3 
AND APPROACH 4,
YOU CAN SEE 3.5, 3.9 AND SO ON,
WHAT WE'RE CONCERNED ABOUT 
IS WHAT'S HAPPENING
TO THE VALUE OF THE FUNCTION 
OR THE Y VALUES.
AND YOU CAN SEE FROM THE TABLE 
THAT IT IS APPROACHING
THE VALUE OF 6.
THIS IS THE IDEA BEHIND A LIMIT.
MATHEMATICALLY 
WE WOULD WRITE THIS
AS THE LIMIT OF S OF X 
AS X APPROACHES 4 IS EQUAL TO 6.
AN INFORMAL DEFINITION, 
AS X APPROACHES C
THE LIMIT OF F OF X IS L 
WRITTEN WITH THIS NOTATION.
IF ALL VALUES OF F OF X 
ARE CLOSE TO L,
FOR VALUES OF X 
THAT ARE SUFFICIENTLY CLOSE
BUT NOT EQUAL TO C.
SUFFICIENTLY CLOSE 
WOULD CONSIST OF VALUES
THAT ARE LESS THAN 
AND GREATER THAN C.
THE VALUES THAT ARE LESS THAN 
WOULD BE TO THE LEFT OF C.
THE VALUES GREATER THAN C 
WOULD BE TO THE RIGHT.
SOME FUNCTIONS MAY NOT HAVE 
LIMITS AT SPECIFIC VALUES OF C.
AND IT'S VERY IMPORTANT TO NOTE 
THAT THE VALUE OF THE LIMIT
IS NOT AFFECTED BY THE VALUE 
OF THE FUNCTION AT C.
LET'S TAKE A LOOK AT A LIMIT, 
THE LIMIT OF F OF X
AS X APPROACHES 1.
WE HAVE THE GRAPH 
AND WE HAVE A TABLE OF VALUES.
LET'S TAKE A LOOK 
AT THE GRAPH FIRST.
ONE THING THAT MIGHT HELP 
IS TO DRAW A VERTICAL LINE
THROUGH X = 1.
SO YOU KIND OF SEE 
WHERE WE'RE APPROACHING.
WE'LL BE APPROACHING 
THIS VERTICAL LINE,
OR YOU CAN THINK OF IT 
AS A WALL,
FROM BOTH THE LEFT SIDE 
AND THE RIGHT SIDE,
AND DETERMINE 
IF WE'RE APPROACHING
THE SAME VALUE.
AGAIN, IT'S IMPORTANT TO NOTE 
THAT THIS FUNCTION
IS NOT EVEN DEFINED AT X = 1, 
BUT IT MAY HAVE A LIMIT THERE.
LET'S TAKE A LOOK.
GRAPHICALLY AS WE APPROACH 
THE VALUE OF X = 1
FROM THE LEFT SIDE IT LOOKS LIKE 
THE Y VALUES ARE APPROACHING -1.
LET'S TAKE A LOOK AT THE T TABLE 
TO SEE IF WE CAN VERIFY THAT.
AS THESE X VALUES APPROACH 1 
FROM THE LEFT WE CAN SEE
THAT THE Y VALUES LOOK LIKE 
THEY ARE APPROACHING -1.
BUT IN ORDER TO DETERMINE 
IF IT HAS A LIMIT THERE,
WE ALSO HAVE TO APPROACH IT 
FROM THE RIGHT SIDE.
IT SEEMS PRETTY CLEAR 
AS WE APPROACH
FROM THE RIGHT SIDE 
USING THIS GREEN ARROW
WE ARE ALSO APPROACHING 
A Y VALUE OF -1.
LET'S LOOK AT THE TABLE AS WELL.
AS WE APPROACH X = 1 
FROM VALUES TO THE RIGHT
OR GREATER THAN 1 HERE, 
IT DOES LOOK LIKE
WE'RE APPROACHING 
A Y VALUE OF -1.
THEREFORE, WE CAN CONCLUDE THAT 
THIS LIMIT WILL BE EQUAL TO -1.
IN GENERAL, THERE ARE THREE WAYS 
TO APPROACH FINDING LIMITS,
THE NUMERICAL APPROACH 
BY USING A T TABLE,
THE GRAPHICAL APPROACH 
BY ANALYZING A GRAPH,
AND THE ANALYTICAL METHOD 
IN WHICH YOU USE
ALGEBRA OR CALCULUS.
THIS WILL BE ADDRESSED 
IN A FUTURE VIDEO.
SINCE WE NOW KNOW ABOUT 
THIS LIMIT NOTATION
AND ROUGHLY WHAT IT MEANS, 
IT IS A GOOD TIME TO INTRODUCE
THE IDEA OF ONE-SIDED LIMITS.
IF YOU TAKE A LOOK AT THE WAY 
THIS LIMIT IS WRITTEN
AS X APPROACHES C, 
AND THEN YOU LOOK
AT THE NOTATION USED DOWN HERE, 
THE ONLY DIFFERENCE
IS THIS ONE HAS A NEGATIVE SIGN 
AND THIS ONE HAS A POSITIVE SIGN
IN THE UPPER 
RIGHT-HAND CORNER OF C.
THIS NOTATION REFERS 
TO THE LIMIT FROM THE LEFT
OR FROM VALUES 
THAT ARE LESS THAN C.
THIS NOTATION REFERS TO A LIMIT 
FROM THE RIGHT OR VALUES
THAT ARE GREATER THAN C.
SO BASICALLY IT'S BREAKING 
THIS LIMIT DOWN
INTO THE LEFT SIDE 
AND RIGHT SIDE.
SO THIS THEOREM FOLLOWS.
AS X APPROACHES C, 
THE LIMIT OF F OF X IS L
IF THE LIMIT 
FROM THE LEFT EXISTS,
AND THE LIMIT 
FROM THE RIGHT EXISTS,
AND THEY'RE BOTH EQUAL TO L.
SO, ESSENTIALLY 
WHAT THIS IS SAYING
IS IF THE LEFT-SIDED LIMIT 
IS EQUAL TO L,
THE RIGHT-SIDED LIMIT 
IS EQUAL TO L,
IT FOLLOWS THAT THE LIMIT 
WILL BE EQUAL TO L.
LET'S TAKE A LOOK AT AN EXAMPLE.
CONSIDER THE FOLLOWING LIMITS.
WE HAVE TWO ONE-SIDED LIMITS 
AND THEN THE LIMIT.
ALL OF THESE ARE APPROACHING 1, 
THEREFORE I'M GOING
TO SKETCH A WALL 
OR A VERTICAL LINE
THROUGH X = 1.
THIS IS SOMETIMES 
CALLED THE WALL METHOD.
THE LIMIT OF F OF X 
AS X APPROACHES 1
FROM THE RIGHT SIDE 
OR POSITIVE SIDE,
GRAPHICALLY THAT 
WOULD REFER APPROACHING 1
ALONG THIS GREEN ARROW.
AS WE CAN SEE 
THAT THE Y VALUES LOOK LIKE
THEY'RE APPROACHING -2.
LOOKING AT THE TABLE 
WITH THE GREEN ARROW,
YOU CAN SEE AS X APPROACHES 1 
FROM THE RIGHT SIDE
IT DOES LOOK LIKE THE Y VALUES 
ARE APPROACHING -2.
THEREFORE, THE LIMIT 
FROM THE RIGHT IS EQUAL TO -2.
NOW, IF WE TAKE A LOOK 
AT THIS LIMIT AS X APPROACHES 1
FROM THE LEFT, ON THE FUNCTION 
WE'D BE APPROACHING
FROM THIS RIGHT ARROW 
AND IT LOOKS LIKE THE Y VALUES
NOW ARE APPROACHING 3.
LET'S VERIFY THAT 
FROM THE TABLE.
AGAIN, AS WE APPROACH 1 
FROM THE LEFT SIDE OR VALUES
THAT ARE LESS THAN ONE, 
THIS DOES VERIFY
IT'S APPROACHING 3.
NOW, REMEMBER 
TO FIND THIS LIMIT,
THESE TWO ONE-SIDED LIMITS 
MUST BE THE SAME,
AND OBVIOUSLY THEY'RE NOT.
THEREFORE, WE WOULD CONCLUDE 
THAT THIS LIMIT DOES NOT EXIST.
LET'S TAKE A LOOK AT ANOTHER.
GIVEN THE FUNCTION H OF X, 
FIND THE LIMITS.
NOW, THE FIRST THING 
I WOULD PROBABLY DO
IS LOOK AT THE GRAPH OF THIS 
AND POSSIBLY MAKE A T TABLE.
LET'S START 
BY LOOKING AT THE GRAPH.
THIS IS THE KIND OF GRAPH 
YOU WOULD GET
FROM YOUR GRAPHING CALCULATOR.
FIRST THING I'M GOING TO DO 
IS DRAW A VERTICAL LINE
THROUGH X = -2
TO KIND OF GET A VISUAL 
OF WHAT WE'RE APPROACHING.
SO WE'RE APPROACHING THAT 
YELLOW WALL FROM THE LEFT SIDE,
THE RIGHT SIDE, 
AND THEN BOTH SIDES.
SO IF I'M APPROACHING -2 FROM 
THE LEFT OR THE NEGATIVE SIDE,
I WOULD BE APPROACHING 
ALONG THIS RED ARROW.
AND WHAT HAPPENS IN THIS CASE 
AS WHEN YOU GET CLOSER
AND CLOSER TO -2, THE GRAPH 
GOES DOWN INDEFINITELY.
THEREFORE, 
THE VALUE OF THE FUNCTION
WOULD BE APPROACHING 
NEGATIVE INFINITY.
FOR THE SECOND ONE-SIDED LIMIT, 
AS X APPROACHES -2
FROM THE RIGHT 
OR THE POSITIVE SIDE,
WE WOULD BE APPROACHING 
FROM THE DIRECTION
OF THIS GREEN ARROW.
SO AS WE APPROACH -2 
FROM THE RIGHT, THE GRAPH
GOES UP INDEFINITELY, 
THEREFORE THE Y VALUES
OR THE FUNCTION VALUES 
WOULD BE APPROACHING
POSITIVE INFINITY.
WELL, OBVIOUSLY 
THESE ONE-SIDED LIMITS
ARE NOT APPROACHING 
THE SAME VALUE,
THEREFORE WE WOULD SAY 
THIS LIMIT DOES NOT EXIST.
NOW, IT'S ALSO IMPORTANT 
TO NOTE THAT INFINITY
AND NEGATIVE INFINITY THEMSELVES 
DON'T EXIST.
SO I MIGHT WANT 
TO NOTE THAT IN HERE.
BUT IT IS COMMON PRACTICE 
TO INCLUDE EITHER INFINITY
OR NEGATIVE INFINITY 
WHEN IT APPLIES
BECAUSE IT DOES GIVE 
ADDITIONAL INFORMATION.
OKAY, I HOPE THAT HELPS 
EXPLAIN A LITTLE BIT
ABOUT THE IDEA OF LIMITS 
AND HOW YOU CAN DETERMINE THEM.
MORE VIDEOS ON LIMITS 
WILL FOLLOW.
THANK YOU FOR TUNING IN 
AND HAVE A WONDERFUL DAY.
