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LIMITS AND CONTINUITY
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
FUNCTION
Definition
Afunctionf is a rule that assigns to each valuex in a setD a unique value denotedf(x).
The setD is the domain of the function.
The set of all values off(x) produced asx varies over
domain is range.
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
FUNCTION
Domain and range
Domain Consisted all values ofx that is possible
Range The set ofy values whenx varies over the domain
Example
Given
2
y x
2y xDomain :
Range :
{ : }D x x R
{ : 0, }R y y y R
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
FUNCTION
Common restriction for function
Type offunction
Example Restriction Remarks
Root
Reciprocal
log or ln
,
2,4,
n a
n
1
a
log ( )
l n ( )
a
a
0a
0a
0a
The function willbecome complex
number if .
Any number dividedby zero is undefined.
0a
log or ln foris undefined.
0a
Example: Domain and range
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
CONCEPT OF LIMITS
The limit of a function is concerned with the behavior of afunction as the independent variable approaches
a given value.If the limit of functionf(x) asx approaches the point a isthe valueL, then it is denoted as
lim ( )x a
f x L
This is not the exact definition of a limit. The definitiongiven above is more of a working definition.
This definition helps us to get an idea of just what limitsare and what they can tell us about functions.
.
In simpler terms, the definition says that as x get closerand closer to a then f(x) must getting closer andcloser to L.
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Consider the following example
Estimate the value of the following limit.2
22
4 12lim
2x
x x
x x
Choose values ofx that get closer to closer to 2 and
substitute the values into . Doing this, gives
the following table:
2
24 12
2x x
x x
x 1.5 1.9 1.999 1.9999 1.99999 2 2.00001 2.0001 2.001 2.01 2.5
f(x) 5.0 4.158 4.002 4.0002 4.00002 3.99999 3.9999 3.999 3.985 3.4
Approach to 2 from the right sideApproach to 2 from the left side
CONCEPT OF LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
The function is going to 4 asxapproaches 2, then
2
22
4 12lim 4.2x
x xx x
Using tables of values to guess the value of limits is simplynot a good way to get the value of a limit but theycan help us get a better understanding of what limits are.
CONCEPT OF LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
One-sided limits: Right side
Only look for limit at right side of the point.
( )f x
a
L
Right-handed
x approaches a from
right hand side.
We write as .lim ( )x a
f x L
CONCEPT OF LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
One-sided limits : Left side
Only look for limit at left side of the point.
( )f x
a
L
Left-handed
lim ( )x a
f x L
x approaches a from
left hand side.
We write as .
CONCEPT OF LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Two-sided limits
>> Requires value of limits from right and left sides.Two-sided limit off(x) is exist if
lim ( ) lim ( )x a x a
f x f x L
Then,
lim ( )x a
f x L
Example: One and Two-sided limits
The limitlim ( )x a
f x L
is called two-sided limit.
CONCEPT OF LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
INFINITE LIMITS
Infinite limits
Infinite limits occur when and .lim ( )x a
f x lim ( )x a f x
All limits that have infinite limit does not exist.
Example: Infinite limits
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
Some basic limits
limx a
k k
Letaandkis any real number
limx a
x a
Consider first limits at a point which islimits off(x) whenx approaches any point a.
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
We will discuss algebraic techniques for computing limitsof:-
Polynomial functions
Rational functions
Radical functions, (function involve )
Piecewise functions
Absolute functions
n
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Polynomial functions
Letf(x) is polynomial function
Then,
lim ( ) ( )x a
f x f a
2
0 1 2( )n
nf x c c x c x c x
Example: Polynomial function
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
, we have to factor d(x) and n(x)(by common factor (xa) ) then cancel out
(SIMPLIFIED THE FUNCTION).
Rational functions
Letf(x) is rational function, with d(x) and n(x) polynomial function
( )( )
( )
n xf x
d x
Then,( )
lim ( ) ( )
( )x a
n af x f a
d a
,provided that .( ) 0d a
But how if ???( ) 0d a
1 If ( ) 0n a lim ( )x af x
does not exist.
2 If( ) 0n a
0
lim ( ) 0x a f x
Example: Rational function
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
ForRationalize is:-
,provided that .
Radical functions, (function involve )n
Letf(x) is radical function, with e(x) and m(x) radical function
( )( )
( )
m xf x
e x
( )
lim ( ) ( )x a
m a
f x e a ( ) 0e a
But how if ???( ) 0e a
1 If
2 If
( ) 0m a lim ( )x a
f x
does not exist.
, rationalizee (x) or m (x)
( ) 0m a 0lim ( )0x a
f x
a b
2
2
2
( )( )a b a b
a b a b a b
a b
Example: Radical function
TECHNIQUES OF COMPUTING LIMITS
Then,Rationalize???
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Piecewise functions
Piecewise function is defined by
1
2
,
,
f x a x b
f x f x b x c
The limits of piecewise function is obtained by finding theone-sided limits first.
Example: Piecewise function
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Absolute functions
Absolute function is defined by ( )f x b x
The limits of absolute function is
lim ( ) ( ) ( )x a
f x f a b a
But how if (A is any number)0( ) or ???0 0Ab a
1 If lim ( )x af x
does not exist.( )
0
Ab a
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
20
( )0
b a If
Example:Absolute function
Consider these notes (a is any real number)
( )
( ) ( )
x a x a
f x x a x a x a
( )( )
( )
a x x af x a x
a x x a
1
2
And always check limits from both sides!
The limit is obtained by finding the one-sided limit first
TECHNIQUES OF COMPUTING LIMITS
( )
x x a
f x x x x a
( )x x a
f x xx x a
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
Some basic limits
limx
k k
Letkis any real number
limx
x
Limits off(x) whenxincrease without bound ( )orxdecrease without bound ( ).
x x
limx
x
1lim 0x x
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
TECHNIQUES OF COMPUTING LIMITS
We will discuss algebraic techniques for computing limitsof:-
Polynomial functions
Rational functions
Radical functions, (function involve )n
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
depends on the highest degree term and
consider
Polynomial functions
lim ( )x
f x
Example: Polynomial function
TECHNIQUES OF COMPUTING LIMITS
Letf(x) is polynomial function
Then,
2
0 1 2( )n
nf x c c x c x c x
, 1,3,5, ( )lim , lim
, 2, 4, 6, ( )
n n
x x
n odd x x
n even
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Letf(x) is rational function( )( )
( )
n xf x
d x
Then,
lim ( )x
f x
Step 1 Divide each term in n(x) and d(x) with
the highest power ofx.
Step 2 Substitute to find the answer
Example: Rational function
Rational functions
TECHNIQUES OF COMPUTING LIMITS
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
Letf(x) is radical function( )
( )( )
m xf x
e x
Case 1 Case 2
f(x) = m (x) only
Step 1 Divide m(x) and e(x) with the highest
power ofxStep 2 Substitute to find the answerx
Rationalize m (x)
Example: Radical function
Radical functions, (function involve )
TECHNIQUES OF COMPUTING LIMITS
n
Try substitute the value of into the function. If cantfind the answer, consider these cases:
( )( )
( )
m xf x
e x
Special case? Use concept2x x
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
CONTINUITY
We perceive the path of moving object as an unbrokencurve without gaps, breaks or holes. In this section,
we translate the unbroken curve into a precisemathematical formulation called continuity.
A functionf(x) is said to be continuous atx = a if
lim ( ) ( )x a
f x f a
Example: Continuity
Provided that EXIST and DEFINED.lim ( )x a
f x
( )f a
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DEPARTMENT OF MATHEMATICS AND STATISTICS, FSTPi
CONTINUITY
Type of discontinuity
Removable discontinuity
Hole
Jump discontinuity
Jump
00lim ( ) ( )
x xf x f x
0 0
lim ( ) lim ( )x x x x
f x f x
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DEPARTMENT OF MATHEMATICS AND STATISTICS FSTPi
Infinite discontinuity
CONTINUITY
Infinite
Example: Discontinuity
0 0lim ( ) or lim ( )x x x xf x f x
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