- WELCOME TO FOUR EXAMPLES OF DETERMINING BASIC LIMITS.
WE FIRST HAVE THE LIMIT AS X APPROACHES -2 OF 4.
SO NOTICE FOR THIS EXAMPLE WE'RE USING
THE CONSTANT FUNCTION F OF X = 4 TO DETERMINE THIS LIMIT.
SO, LET'S BEGIN BY LOOKING AT THE GRAPH
OF THIS CONSTANT FUNCTION.
IT'S A HORIZONTAL LINE PASSING THROUGH Y = 4.
AND HERE'S WHERE X = -2.
AND NOTICE AS WE APPROACH -2 FROM THE POSITIVE SIDE
OR RIGHT SIDE, AND AS WE APPROACH -2
FROM THE LEFT SIDE OR NEGATIVE SIDE,
NOT ONLY IS THE FUNCTION VALUE APPROACHING 4, IT'S ALWAYS 4
BECAUSE WE HAVE THE CONSTANT FUNCTION.
AND, THEREFORE, THIS LIMIT IS EQUAL TO 4.
BUT THIS DOES BRING UP AN IMPORTANT POINT
WHEN DETERMINING LIMITS.
THE LIMIT AS X APPROACHES C OF F OF X = F OF C
MEANING WE CAN DETERMINE THIS LIMIT BY PERFORMING
DIRECT SUBSTITUTION WITH X = C IF F OF X IS CONTINUOUS
OVER AN OPEN INTERVAL FROM A TO B THAT CONTAINS C,
WHICH MEANS WE SHOULD BECOME FAMILIAR
WITH THE TYPES OF LIMITS THAT CAN BE DETERMINED
USING DIRECT SUBSTITUTION BY KNOWING WHERE FUNCTIONS
ARE CONTINUOUS AND NOT CONTINUOUS.
AND IF A FUNCTION IS CONTINUOUS OVER ALL REAL NUMBERS,
WE CAN ALWAYS DETERMINE THE LIMIT BY PERFORMING
DIRECT SUBSTITUTION.
SO, LOOKING AT OUR SECOND EXAMPLE,
WE HAVE THE LIMIT AS X APPROACHES 3 OF X,
WHICH MEANS, AGAIN, OUR FUNCTION IS F OF X = X GRAPHED HERE,
WHICH WOULD BE THE IDENTITY FUNCTION.
BUT BECAUSE THIS IS A LINE, WHICH IS ALWAYS CONTINUOUS
OVER ALL REAL NUMBERS, WE CAN SIMPLY PERFORM
DIRECT SUBSTITUTION TO DETERMINE THIS LIMIT.
SO, IF WE SUBSTITUTE 3 FOR X, NOTICE HOW THE LIMIT
IS JUST EQUAL TO 3.
TO UNDERSTAND WHAT'S HAPPENING GRAPHICALLY,
HERE'S WHERE X = 3.
AS WE APPROACH X = 3 FROM THE RIGHT SIDE OR POSITIVE SIDE,
AND AS WE APPROACH 3 FROM THE LEFT SIDE
OR NEGATIVE SIDE,
IN BOTH CASES WE ARE APPROACHING THE FUNCTION VALUE OF 3.
NEXT, WE HAVE THE LIMIT AS X APPROACHES 4 OF -2X + 3.
ONCE AGAIN, NOTICE THE FUNCTION IS A LINE WHERE WE HAVE
F OF X = -2X + 3, WHICH, AGAIN, IS CONTINUOUS
OVER ALL REAL NUMBERS.
AND, THEREFORE, WE CAN DETERMINE THIS LIMIT
BY PERFORMING DIRECT SUBSTITUTION,
WHICH WOULD JUST BE -2 x 4 + 3,
WHICH WOULD BE -8 + 3 OR -5.
SO, THE LIMIT AS X APPROACHES 4 OF -2X + 3 = -5.
WE'RE ANALYZING THE GRAPH. HERE'S WHERE X = 4.
NOTICE AS WE APPROACH 4 FROM THE RIGHT SIDE
OR POSITIVE SIDE, AND AS WE APPROACH 4
FROM THE LEFT SIDE OR NEGATIVE SIDE,
NOTICE FROM BOTH DIRECTIONS WE'RE APPROACHING
THE FUNCTION VALUE -5.
AND FOR OUR LAST EXAMPLE WE HAVE THE LIMIT
AS X APPROACHES 0 OF 0.5X CUBED + X SQUARED,
NOTICE HOW OUR FUNCTION IS POLYNOMIAL FUNCTION, WHICH,
ONCE AGAIN, IS CONTINUOUS OVER ALL REAL NUMBERS.
AND THEREFORE WE CAN FIND THIS LIMIT
BY PERFORMING DIRECT SUBSTITUTION.
WE SUBSTITUTE 0 FOR X, WE WOULD HAVE 0.5 x 0 CUBED
+ 0 SQUARED, WHICH WOULD JUST BE 0 + 0 OR 0.
LOOKING AT THE GRAPH, HERE'S WHERE X = 0.
AS WE APPROACH X = 0 FROM THE POSITIVE SIDE OR RIGHT SIDE,
AND AS WE APPROACH 0 FROM THE LEFT SIDE OR NEGATIVE SIDE,
NOTICE HOW FROM BOTH DIRECTIONS WE ARE APPROACHING
THE FUNCTION VALUE F OF X OR Y = 0.
I HOPE YOU HAVE FOUND THESE FOUR EXAMPLES HELPFUL.