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Limits and Continuity
Definition
Evaluation of LimitsContinuity
Limits Involving Infinity
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Limit of the Function
Note: we can approach a limit from
left right both sides
Function may or may not exist at that point At a
right hand limit, no left
function not defined
At b left handed limit, no right
function defined
a b
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Can be observed on a graph.
Observing a Limit
View
Demo
http://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits2.ggb8/22/2019 2Limits and Continuity (1)
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Observing a Limit
Can be observed on a graph.
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Observing a Limit
Can be observed in a table
The limit is observed to be 64
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Non Existent Limits
Limits may not exist at a specific point for a
function
Set
Consider the function as it approaches
x = 0
Try the tables with start at0.03, dt = 0.01
What results do you note?
11( )
2y x
x
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Non Existent Limits
Note that f(x) does NOT get closer to a
particular value
it grows without bound
There is NO LIMIT
Try command on
calculator
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Non Existent Limits
f(x) grows without bound
View
Demo3
http://localhost/var/www/apps/conversion/tmp/scratch_15/limits3.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits3.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits3.ggb8/22/2019 2Limits and Continuity (1)
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Non Existent Limits
View
Demo 4
http://localhost/var/www/apps/conversion/tmp/scratch_15/limits4.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits4.ggbhttp://localhost/var/www/apps/conversion/tmp/scratch_15/limits4.ggb8/22/2019 2Limits and Continuity (1)
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Formal Definition of a Limit
The
For any (as close as
you want to get to L) There exists a (we can get as close as
necessary to c )
lim ( )x c
f x L
L
c
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Formal Definition of a Limit
For any (as close as you want to get to L)
There exists a (we can get as close as
necessary to c
Such that
( )f x L when x c
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Specified Epsilon, Required Delta
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Limit
We say that the limit of ( ) as approaches is and writef x x a L
lim ( )x a
f x L
if the values of ( ) approach as approaches .f x L x a
a
L
( )y f x
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c) Find2
3 if 2lim ( ) where ( )1 if 2x
x xf x f xx
-2
62 2
lim ( ) = lim 3x x
f x x
Note: f(-2) = 1
is not involved
23 lim
3( 2) 6
xx
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2
24( 4)a. lim
2xx
x
0
1, if 0b. lim ( ), where ( )
1, if 0x
xg x g x
x
20
1c. lim ( ), where f ( )
xf x x
x
0
1 1d. lim
x
x
x
Answer : 16
Answer : no limit
Answer : no limit
Answer : 1/2
3) Use your calculator to evaluate the limits
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The Definition of Limit-
lim ( )We say if and only ifx a
f x L
given a positive number , there exists a positive such that
if 0 | | , then | ( ) | .x a f x L
( )y f xa
LL
L
a a
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such that for all in ( , ),x a a a
then we can find a (small) interval ( , )a a
( ) is in ( , ).f x L L
This means that if we are given a
small interval ( , ) centered at ,L L L
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Examples
21. Show that lim(3 4) 10.
xx
Let 0 be given.
We need to find a 0 such that
if | - 2 | ,x then | (3 4) 10 | .x
But | (3 4) 10 | | 3 6 | 3 | 2 |x x x
if | 2 |3
x
So we choose .3
1
12. Show that lim 1.
x x
Let 0 be given. We need to find a 0 such that 1if | 1| , then | 1| .x
x
1 11But | 1| | | | 1| .x
xx x x
What do we do with the
x?
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1 31If we decide | 1| , then .2 22
x x
1And so
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The right-hand limit off(x), asx approaches a,equalsL
written:
if we can make the valuef(x) arbitrarily close
toLby takingx to be sufficiently close to the
right ofa.
lim ( )x a
f x L
a
L
( )y f x
One-Sided Limit
One-Sided Limits
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The left-hand limit off(x), asx approaches a,
equalsM
written:
if we can make the valuef(x) arbitrarily close
toLby takingx to be sufficiently close to theleft ofa.
lim ( )x a
f x M
a
M
( )y f x
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2
if 3( )2 if 3x xf x
x x
1. Given
3lim ( )
xf x
3 3lim ( ) lim 2 6
x xf x x
2
3 3lim ( ) lim 9
x xf x x
Find
Find3
lim ( )x
f x
Examples
Examples of One-Sided Limit
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1, if 02. Let ( )
1, if 0.
x xf x
x x
Find the limits:
0lim( 1)
xx
0 1 1
0a) lim ( )
xf x
0b) lim ( )
xf x 0
lim( 1)x
x
0 1 1
1
c) lim ( )x
f x 1
lim( 1)x
x
1 1 2
1d) lim ( )
xf x
1lim( 1)
xx
1 1 2
More Examples
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lim ( ) if and only if lim ( ) and lim ( ) .x a x a x a
f x L f x L f x L
For the function
1 1 1lim ( ) 2 because lim ( ) 2 and lim ( ) 2.x x x
f x f x f x
But
0 0 0lim ( ) does not exist because lim ( ) 1 and lim ( ) 1.x x xf x f x f x
1, if 0( )
1, if 0.
x xf x
x x
This theorem is used to show a limit does not
exist.
A Theorem
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Limit Theorems
If is any number, lim ( ) and lim ( ) , thenx a x a
c f x L g x M
a) lim ( ) ( )x a
f x g x L M
b) lim ( ) ( )x a
f x g x L M
c) lim ( ) ( )x a
f x g x L M
( )d) lim , ( 0)( )x a f x L Mg x M
e) lim ( )x a
c f x c L
f) lim ( )n n
x af x L
g) limx a
c c h) limx a
x a
i) lim n nx a
x a
j) lim ( ) , ( 0)x a
f x L L
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Examples Using Limit Rule
Ex. 23
lim 1
x
x
2
3 3
lim lim1
x x
x
2
3 3
2
lim lim1
3 1 10
x xx
Ex.1
2 1lim
3 5x
x
x
1
1
lim 2 1
lim 3 5
x
x
x
x
1 1
1 1
2lim lim1
3lim lim5
x x
x x
x
x
2 1 1
3 5 8
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More Examples
3 3
1. Suppose lim ( ) 4 and lim ( ) 2. Findx x
f x g x
3
a) lim ( ) ( )x
f x g x
3 3
lim ( ) lim ( )x x
f x g x
4 ( 2) 2
3
b) lim ( ) ( )x
f x g x
3 3
lim ( ) lim ( )x x
f x g x
4 ( 2) 6
3
2 ( ) ( )c) lim
( ) ( )x
f x g x
f x g x
3 3
3 3
lim 2 ( ) lim ( )
lim ( ) lim ( )
x x
x x
f x g x
f x g x
2 4 ( 2) 5
4 ( 2) 4
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Indeterminate forms occur when substitution in the limit
results in 0/0. In such cases either factor or rationalize the
expressions.
Ex.25
5lim
25x
x
x
Notice form00
5
5lim
5 5x
x
x x
Factor and cancel
common factors
51 1
lim5 10x x
Indeterminate Forms
M E l
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9
3a) lim
9x
x
x
9
( 3)
( 3)
( 3)= lim
( 9)x
x
x
x
x
9
9lim
( 9)( 3)x
x
x x
9
1 1lim
63x x
2
2 32
4b) lim
2x
x
x x
22
(2 )(2 )= lim
(2 )x
x x
x x
222= lim
x
x
x
2
2 ( 2) 41
( 2) 4
More Examples
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Co ri ht c 2003 Brooks/Cole a division of
Computing Limits
Ex.2
3 if 2lim ( ) where ( )1 if 2x
x xf x f xx
6
-2
2 2lim ( ) = lim 3
x xf x x
23 lim
3( 2) 6
xx
Note: f(-2) = 1
is not involved
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Co ri ht c 2003 Brooks/Cole a division of
Computing LimitsEx.
Ex.
23
lim 1
x
x
2
3 3
lim lim1
x x
x
2
3 3
2
lim lim1
3 1 10
x xx
1
2 1lim
3 5x
x
x
1
1
lim 2 1
lim 3 5
x
x
x
x
1 1
1 1
2lim lim1
3lim lim5
x x
x x
x
x
2 1 1
3 5 8
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Co ri ht c 2003 Brooks/Cole a division of
Indeterminate Forms:0
0
25
5lim
25x
x
x
Ex. Notice form
0
0
55
lim 5 5x
x
x x
51 1
lim5 10x x
Factor and cancelcommon factors
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Co ri ht c 2003 Brooks/Cole a division of
Limits at Infinity
For all n > 0,1 1lim lim 0n nx xx x
provided that is defined.1
nx
Ex.2
2
3 5 1lim
2 4x
x x
x
2
2
5 13lim
2 4x
x x
x
3 0 0 3
0 4 4
Divide
by 2x
2
2
5 1lim 3 lim lim
2lim lim 4
x x x
x x
x x
x
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Co ri ht c 2003 Brooks/Cole a division of
One-Sided Limit of a Function
The right-hand limit off(x), asx approaches a, equalsL
written:
if we can make the valuef(x) arbitrarily close toLbytakingx to be sufficiently close to the right ofa.
lim ( )x a
f x L
a
L
( )y f x
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Co ri ht c 2003 Brooks/Cole a division of
One-Sided Limit of a Function
The left-hand limit off(x), asx approaches a, equalsM
written:
if we can make the valuef(x) arbitrarily close toLbytakingx to be sufficiently close to the left ofa.
lim ( )x a
f x M
a
M
( )y f x
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One-Sided Limit of a Function2
if 3( )2 if 3x xf x
x x
Ex. Given
3lim ( )
xf x
3 3lim ( ) lim 2 6
x xf x x
2
3 3lim ( ) lim 9
x xf x x
Find
Find
3
lim ( )x
f x
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Continuity of a Function
A functionfis continuousat the pointx = a if thefollowing are true:
) ( ) is definedi f a
) lim ( ) existsx aii f x
a
f(a)
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Continuity
A functionfis continuousat the pointx = a ifthe following are true:
) ( ) is definedi f a
) lim ( ) existsx aii f x
a
f(a)
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A functionfis continuousat the pointx = a ifthe following are true:
) ( ) is definedi f a
) lim ( ) existsx aii f x
) lim ( ) ( )x a
iii f x f a
a
f(a)
E l
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At which value(s) of x is the given function
discontinuous?
1. ( ) 2f x x
2
92. ( )3
xg xx
Continuous everywhereContinuous everywhere
except at3x
( 3) is undefinedg
lim( 2) 2x a
x a
and so lim ( ) ( )x a
f x f a
-4 -2 2 4
-2
2
4
6
-6 -4 -2 2 4
-10
-8
-6
-4
-2
2
4
Examples
1 if 0
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2, if 13. ( )
1, if 1
x xh x
x
1lim ( )x h x and
Thus h is not cont. atx=1.
1 1lim ( )x h x 3
h is continuous everywhere else
1, if 04. ( )
1, if 0
xF x
x
0
lim ( )x
F x
1and
0
lim ( )x
F x
1
ThusFis not cont. at 0.x
F is continuous everywhere else
-2 2 4
-3
-2
-1
1
2
3
4
5
-10 -5 5 10
-3
-2
-1
1
2
3
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Continuous Functions
Apolynomial functiony =P(x) is continuous at
every pointx.
A rational function is continuous
at every pointx in its domain.
( )( )
( )p x
R xq x
Iffandgare continuous atx = a, then
, , and ( ) 0 are continuous
at
ff g fg g a
g
x a
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Intermediate Value Theorem
Iffis a continuous function on a closed interval [a, b]andL is any number betweenf(a) andf(b), then there
is at least one numberc in [a, b] such thatf(c) =L.
( )y f x
a b
f(a)
f(b)
L
c
f(c) =
Example
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Example
2Given ( ) 3 2 5,
Show that ( ) 0 has a solution on 1,2 .
f x x x
f x
(1) 4 0(2) 3 0
ff
f(x) is continuous (polynomial) and sincef(1) < 0andf(2) > 0, by the Intermediate Value Theorem
there exists a c on [1, 2] such thatf(c) = 0.
Li it t I fi it
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Limits at Infinity
For all n > 0, 1 1lim lim 0n nx xx x
provided that is defined.1
nx
Ex.2
2
3 5 1lim
2 4x
x x
x
2
2
5 13lim
2 4x
x x
x
3 0 0 3
0 4 4
Divide
by2x
2
2
5 1lim 3 lim lim
2lim lim 4
x x x
x x
x x
x
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More Examples
3 2
3 2
2 3 21. lim
100 1x
x x
x x x
3 2
3 3 3
3 2
3 3 3 3
2 3 2
lim 100 1x
x x
x x x
x x x
x x x x
3
2 3
3 22
lim1 100 1
1x
x x
x x x
22
1
2
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0
2
3 2
4 5 212. lim
7 5 10 1x
x x
x x x
2
3 3 3
3 2
3 3 3 3
4 5 21
lim7 5 10 1x
x x
x x x
x x x
x x x x
2 3
2 3
4 5 21
lim5 10 1
7x
x x x
x x x
07
2 2 43. lim
12 31x
x x
x
2 2 4
lim12 31x
x x
x x xx
x x
42
lim31
12x
xx
x
2
12
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24. lim 1x
x x
2 22
1 1lim
1 1x
x x x x
x x
2 2
2
1lim
1xx x
x x
2
1lim
1x x x
1 10
I fi i Li i
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Infinite LimitsFor all n > 0,
1lim
nx a x a
1lim if is even
nx a
nx a
1lim if is odd
nx a
nx a
-8 -6 -4 -2 2
-20
-15
-10
-5
5
10
15
20
-2 2 4 6
-20
-10
10
20
30
40
More Graphs
-8 -6 -4 -2 2
-15
-10
-5
5
10
15
20
http://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_15/infinitelimit/index.html8/22/2019 2Limits and Continuity (1)
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Examples
Find the limits
2
20
3 2 11. lim
2x
x x
x
2
0
2 13= lim
2x
x x
3
2
3
2 12. lim2 6x
xx
3
2 1= lim
2( 3)x
x
x
-8 -6 -4 -2 2
-20
20
40
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Limit and Trig Functions
From the graph of trigs functions
( ) sin and ( ) cosf x x g x x
we conclude that they are continuous everywhere
-10 -5 5 10
-1
-0.5
0.5
1
-10 -5 5 10
-1
-0.5
0.5
1
limsin sin and limcos cosx c x c
x c x c
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Tangent and SecantTangent and secant are continuous everywhere in their
domain, which is the set of all real numbers
3 5 7, , , ,2 2 2 2
x
-6 -4 -2 2 4 6
-30
-20
-10
10
20
30
-6 -4 -2 2 4 6
-15
-10
-5
5
10
15
tany x
secy x
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Examples
2a) lim sec
x
x
2
b) lim secx
x
3 2c) lim tan
x
x
3 2
d) lim tanx
x
e) lim cotx
x
3 2g) lim cot
x
x
3 2
cos 0lim 0
sin 1x
x
x
4
f) lim tanx
x
1
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Limit and Exponential Functions
-6 -4 -2 2 4 6
-2
2
4
6
8
10
, 1xy a a
-6 -4 -2 2 4 6
-2
2
4
6
8
10
, 0 1xy a a
The above graph confirm that exponential
functions are continuous everywhere.
lim x cx c
a a
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Asymptotes
horizontal asymptotThe line is called a
of the curve ( ) if eihter
ey L
y f x
lim ( ) or lim ( ) .x xf x L f x L
vertical asymptoteThe line is called a
of the curve ( ) if eihter
x c
y f x
lim ( ) or lim ( ) .x c x c
f x f x
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Examples
Find the asymptotes of the graphs of the functions2
2
11. ( )
1
xf x
x
1(i) lim ( )
xf x
Therefore the line 1
is a vertical asymptote.
x
1.(iii) lim ( )x
f x
1(ii) lim ( )
xf x
.
Therefore the line 1
is a vertical asymptote.
x
Therefore the line 1
is a horizonatl asymptote.
y
-4 -2 2 4
-10
-7.5
-5
-2.5
2.5
5
7.5
10
1
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2
12. ( )
1
xf x
x
21 1
1(i) lim ( ) lim1x x
xf xx
1 1
1 1 1=lim lim .
( 1)( 1) 1 2x x
x
x x x
Therefore the line 1
is a vertical asympNO t eT ot .
x
1(ii) lim ( ) .
xf x
Therefore the line 1
is a vertical asymptote
x
(iii) lim ( ) 0.x f x Therefore the line 0
is a horizonatl asymptote.
y
-4 -2 2 4
-10
-7.5
-5
-2.5
2.5
5
7.5
10
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