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Teaching Limits so that
Students will Understand Limits
Presented by
Lin McMullinNational Math and Science
Initiative
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( )3 24 6
2
x x x f x
x
− + +=
−
−4 −3 −2 −1 1 2 3 4
−4
−3
−2
−1
1
2
3
4
Continuity
What happens at x = 2?
What is f (2)?
What happens near x = 2?
f ( x ) is near 3
What happens as x approaches 2?
f ( x ) approaches 3
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What happens at x = 1?
What happens near x = 1?
As x approaches 1 g increases !itho"t bo"nd or g approaches in#nity$
As x increases !itho"t bo"nd g approaches %$
As x approaches in#nity g approaches %$
Asymptotes
−4 −3 −2 −1 1 2 3 4 5 6
−1
1
2
3
4
5
6
( )( )
2
1
1 g x
x=
−
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Asymptotes
−4 −3 −2 −1 1 2 3 4 5 6
−1
1
2
3
4
5
6
( )( )
2
1
1 g x
x=
−
& &'1 1(&'1)2
0.9 -0.1 100.00
0.91 -0.09 123.46
0.92 -0.08 156.25
0.93 -0.07 204.08
0.94 -0.06 277.78
0.95 -0.05 400.00
0.96 -0.04 625.00
0.97 -0.03 1,111.11
0.98 -0.02 2,500.00
0.99 -0.01 10,000.00
1 0 Undefned
1.01 0.01 10,000.00
1.02 0.02 2,500.00
1.03 0.03 1,111.11
1.04 0.04 625.00
1.05 0.05 400.00
1.06 0.06 277.78
1.07 0.07 204.08
1.08 0.08 156.25
1.09 0.09 123.46
1.10 0.1 100.00
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Asymptotes
−4 −3 −2 −1 1 2 3 4 5 6
−1
1
2
3
4
5
6
( )( )
2
1
1 g x
x=
−
& &'1 1(&'1)2
1 0 Undefnned
2 1 1
5 4 0.25
10 9 0.01234567901234570
50 49 0.00041649312786339
100 99 0.00010203040506071
500 499 0.00000401604812832
1,000 999 0.00000100200300401
10,000 9999 0.00000001000200030
100,000 99999 0.00000000010000200
1,000,000 999999 0.00000000000100000
10,000,000 9999999 0.00000000000001000
100,000,000 99999999 0.00000000000000010
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1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
The Area Problem
( )
( )
21
0
1
3
h x x
j x
x
x
= +
=
=
=
What is the area o* the o"tlined re+ion?
As the n",ber o* rectan+les increases !ith o"t bo"ndthe area o* the
rectan+les approaches the area o* the re+ion$
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
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The Tangent Line Problem
What is the slope o* the blac- line?
As the red point approaches the blac- point the redsecant line approaches the blac- tan+ent line and
.he slope o* the secant line approaches the slope o* thetan+ent line$
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As x approaches 1, ! " #x $ approaches %
f ( x ) wi t h i n 0 . 0 8
uni t s
o
f 3 xwi t h i n 0 . 0 4 un
i t s
of 1
f (
x ) wi t h i n 0 .1 6 u
ni t s
of 3
x
wi t h i n 0 . 0 8 uni t s
o
f 1
0.90 3.20
0.91 3.18
0.92 3.16
0.93 3.14
0.94 3.12
0.95 3.10
0.96 3.08
0.97 3.06
0.98 3.04
0.99 3.02
1.00 3.00
1.01 2.98
1.02 2.96
1.03 2.94
1.04 2.92
1.05 2.90
1.06 2.88
1.07 2.86
1.08 2.84
1.09 2.82
x 5 2 x−
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( )→
− =1
lim 5 2 3x
x
( )
( )
12
2 2
2 2
5 2 3
5 2 3
x
x
x
x
x
f x L
ε
ε
ε
ε
ε
ε
− <
− <
− <
− − <
− − <
− <
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−2 −1 1 2 3
−1
1
2
3
4
5
1 or 1 12 2 2
x xε ε ε
− < − < < +
( )5 2 3
or
3 5 2 3
x
x
ε
ε ε
− − <
− < − < +
( )→
− =1
lim 5 2 3x
x
Gr!h
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( )→
− =1
lim 5 2 3x
x
2
ε δ = δ ε =
4
ε δ =
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"hen the #$ues su%%essi#e$&ttri'uted to #ri'$e !!ro%h
indefnite$& to fed #$ue, in nner so s to end '& di*erin+ro it s $itt$e s one wishes,
this $st is %$$ed the $iit o $$the others.
u+ustin-ouis
u%h& (1789 / 1857)
The &e'inition o' Limit at a Point
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( )
( )
( ) ( )
lim if, and only if, for any number 0
there is a number 0 such
if 0 ,
t
then
hat
x a
x
L
f x L
f
a
x ε
ε
ε
ε
δ
δ
→
< −
>
< −
=
<
>
The &e'inition o' Limit at a Point
r$ "eierstrss
(1815 / 1897)
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( )
( )
( ) ( )
lim if, and only if, for any number 0
there is a number 0 such
if 0 ,
t
then
hat
x a
x
L
f x L
f
a
x ε
ε
ε
ε
δ
δ
→
< −
>
< −
=
<
>
( )
( )( ) ( ) ( )
lim if, and only if, for any number 0
there is a numb
i
er 0 and such t
f , then
hat
x a
a x a L f x L
f x L
x a
δ ε δ
ε
ε
δ ε
ε ε
→= >
− < < + − < <
≠
+
>
The &e'inition o' Limit at a Point
r$ "eierstrss
(1815 / 1897)
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( )( )
lim 0 0 such that
, whenever 0
x a f x L
f x L x a
ε δ
ε δ
→ = ⇔ ∀ > ∃ >
− < < − <
(ootnote)The &e'inition o' Limit at a Point
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(ootnote)The &e'inition o' Limit at a Point
( )( )
lim 0 0 such
whe
that
, 0never
x a f x L
f x L x a
ε δ
ε δ
→ ⇔ ∀ ∃= > >
− < < − <
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f
( x ) wi t h i n 0 . 0 8
uni t s
o
f 3
xwi t h i n 0 . 0 4 un
i t s
of 1
f (
x ) wi t h i n 0 .1 6 u
ni t s
of 3
x
wi t h i n 0 . 0 8 uni
t s
o
f 1
0.90 3.20
0.91 3.18
0.92 3.16
0.93 3.14
0.94 3.12
0.95 3.10
0.96 3.08
0.97 3.06
0.98 3.04
0.99 3.02
1.00 3.00
1.01 2.98
1.02 2.96
1.03 2.94
1.04 2.92
1.05 2.90
1.06 2.88
1.07 2.86
1.08 2.84
1.09 2.82
x 5 2 x−
( )→
− =1
lim 5 2 3x
x
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( )→
− =1
lim 5 2 3x
x
( )
( )5 2 3
5 2 3
2 2
2 2
12
f x L
x
x
x
x
x
ε
ε
ε
ε
ε
ε
− <
− − <
− − <
− <
− <
− <
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→
=2
3lim 9x
x
2 9
3 3
x
x x
ε
ε
− <
+ − <
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→
=2
3lim 9x
x
2 9
3 3
x
x x
ε
ε
− <
+ − <
Near 3, specifically in 2,4!, 5 3 " x x= < + <
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→
=2
3lim 9x
x
2 9
3 3
x
x x
ε
ε
− <
+ − <
Near 3, specifically in 2,4!, 5 3 " x x= < + <
" 3
3 "
x
x
ε
ε
− <
− <
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→
=2
3lim 9x
x
2 9
3 3
x
x x
ε
ε
− <
+ − <
Near 3, specifically in 2,4!, 5 3 " x x= < + <
" 3
3 "
x
x
ε
ε
− <
− <
( ) the smaller of 1 and"
ε δ ε = Gr!h
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→
=lim sin( ) sin( )x a
x a
Gr!h1
1
a + delta
a - delta
sin x
sin a
(cos x, sin x)
(cos a, sin a)
1, 0!
x in radians
0,1!
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( )
( )
( ) ( )
( )
( )
( )
lim if, and only if, for any number 0
there is a number 0 such that
if 0 , then
lim if, and only if, for any number 0
there is a number 0 such that
if 0 , t
a
a
x
x
f x L
f x x a L
f x L
a x
ε
δ ε
δ ε ε
ε
δ ε
δ ε
+
−
→
→
−
= >
>
< <
−
− <
= >
>
< < ( )hen f x L ε − <
*ne+sided Limits
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Limits -ual to .n'inity
( )
( )
( ) ( ) ( )( )
lim if, and only if, for any number
there is a number such that
if
0
/raphically this is a vertical asymptote
0 , then
x a M
f x M M
f x
x a f x
δ ε
δ ε
→ =
<
>
( )( )
( ) ( )
lim if, and only if, for any number
there is a number such that
if 0 , then
0 x a f x
x
M
f xa M
δ ε
δ ε
→ =
< − <
−∞ <
<
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Limit as x Approaches .n'inity
( )
( )
lim if, and only if, for any number 0
there is a number suc
/raphically, this is a horiontal a
h that
if , then
sym t
0
p ote
x f x L
M f x L
M
x
ε
ε
∞→= >
−> <
>
( )
( )
lim if, and only if, for any number 0there is a number such that
if , then
0
x
M
x
f x L
f x L M
ε
ε
→−∞
<
<
= >
− <
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Limit Theorems
$ost $$ $iit re %tu$$& ound '&su'stitutin+ the #$ues into the e!ression,si!$i&in+, nd %oin+ u! with nu'er,
the $iit.
he theores on $iits o sus, !rodu%ts,!owers, et%. usti& the su'stitutin+.
hose tht dont si!$i& %n oten 'e oundwith ore d#n%ed theores su%h s!it$s u$e
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The Area Problem
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
( )
( )
21
0
1
3
h x x
j x
x
x
= +
=
=
=
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The Area Problem
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
( ) ( )
( ) ( ) ( ){ }
( )( ) ( )
( )( ) ( )
2
2 2 2 2
22 2
1
2
2 2
1
len$th 1
3 1 2width
)*coordinates ' 1,1 ,1 2 ,1 3 , ,1
rea 1 1
32rea ' lim 1 1 3
n n n n
n
n n
i
n
n nn
i
h x j x x
b a
n n n
n
i
i
=
→∞=
= − = +
− −= = =
+ + + +
≈ + +
+ + =
∑
∑
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The Area Problem
1 2 3 4−1
1
2
3
4
5
6
7
8
9
10
( )( ) ( )
( ) ( ) ( )
2
2 3
2 3
2 22 2 2 2 4 2 4
1 1 1 1
4 2
1
1 2 1
6
1
4
3
32
1
3
2
1
lim 1 1 lim 2 lim lim
lim lim lim
lim lim lim
4 4
1
n n n n
n n n n n n ni n n ni i i i
n n nn n n
n n n
n
i
n
n n n
n
n n
i
n
n
n
i
i
i i
n
i
i
→∞ →∞ →∞ →∞= = = =
→∞ →∞ →∞
→∞ →∞ →
=
+
==
+
∞
+
+ + = + +
= + +
= + +
= + +
=
∑∑
∑
∑
∑ ∑ ∑
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( )( ), P a x f a x+ ∆ + ∆
( )( ),T a f a
x∆
( ) ( ) f a x f a+ ∆ −
( ) ( ) y f a x am= + −
The Tangent Line Problem
( ) ( )
( )
s
0
slope
P T
x
PT
f a x f a
a x a
m
m
→
∆ →
→
+ ∆ −→
+ ∆ −
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( )( ), P a x f a x+ ∆ + ∆
( )( ),T a f a
x∆
( ) ( ) f a x f a+ ∆ −
( ) ( ) ( )a y f a x a f ′= + −
The Tangent Line Problem
( ) ( )
( ) ( )
s
0
slope m
P T
x
PT
f f
a x f a
a xa
a
→
∆ →
→
−→
+ ∆ −′
+ ∆
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in %u$$intion$ th nd :%ien%e;nititi#e
325 orth :t.
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0in McM"llin
l,c,"llin/NationalMathAndScience$or+
!!!$0inMcM"llin$net lic- AP
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( ) ( )
( ) ( )
( ) ( )
#et
sin sin
sin sin
limsin sin x a
x a x a
x a x a
x a
δ ε
δ ε
→
=
− < −
− < ⇒ − <
⇒ =
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−2 −1 1 2 3
−1
1
2
3
4
5
( ) x a δ ε − <
( ) f x L ε − <
( )→
− =1
lim 5 2 3x
x
( )
( )
( ) ( )
lim if, and only if,
for any number 0 there is a
number 0 such that if
0 , then
x a f x L
x a f x L
ε
δ ε
δ ε ε
→=
>
>
< − < − <
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( )( ), P a x f a x+ ∆ + ∆
( )( ),T a f a
x∆
( ) ( ) f a x f a+ ∆ −
( ) ( ) y f a x am= + −
The Tangent Line Problem
( ) ( )
( )
s
0
slope
P T
x
PT
f a x f a
a x a
m
m
→
∆ →
→
+ ∆ −→
+ ∆ −