Sec 2.4: The Precise Definition of a Limit
Two positive small number


(the Greek letter delta)
(the Greek letter epsilon)
Sec 2.4: The Precise Definition of a Limit
x  2  1.5
4
It is an interval centered
at 2 with radius 1.5
2
 (2  1.5,2  1.5)
 (0.5,3.5)
4
2
2
2
1.5
4
0  x  2  1.5
1.5
It is an interval centered
at 2 with radius 1.5
without the center {2}
4
 (0.5,3.5)  {2}
 (0.5,2)  (2,3.5)
Sec 2.4: The Precise Definition of a Limit
x  3  0.5
4
It is an interval centered
at 3 with radius 0.5
2
 (3  0.5,3  0.5)
 (2.5,3.5)
4
2
2
4
0  x  3  0.5
2
It is an interval centered
at 3 with radius 0.5
without the center {3}
4
 (2.5,3.5)  {3}
 (2.5,3)  (3,3.5)
Sec 2.4: The Precise Definition of a Limit
xa 
4
delta-interval
2
It is an interval centered
at a with radius

 (a   , a   )
4
2
2
2
4
a

0 xa 
It is an interval centered
at a with radius
without the center

4
 (a   , a   )  {a}
 ( a   , a )  ( a, a   )
Sec 2.4: The Precise Definition of a Limit
y  2  1.5
4
It is an interval along the
y-axis centered at 2 with
radius 1.5
2
 (2  1.5,2  1.5)
4
2
2
2
4
4
 (0.5,3.5)
y  (0.5,3.5)
Sec 2.4: The Precise Definition of a Limit
f ( x)  2  1.5
4
2
4
2
It is an interval along the
y-axis centered at 2 with
radius 1.5
2
2
4
4
f ( x)  (0.5,3.5)
Sec 2.4: The Precise Definition of a Limit
f ( x)  L  

epsilon-interval
L

It is an interval centered
at L with radius

 (L   , L   )
f ( x)  ( L   , L   )
Sec 2.4: The Precise Definition of a Limit
f ( x)  L  

epsilon-interval
It is an interval centered
at L with radius

 (L   , L   )
L

4
2
f ( x)  ( L   , L   )
2
4
a

delta-interval
xa 
It is an interval centered
at a with radius

 (a   , a   )
Sec 2.4: The Precise Definition of a Limit
f ( x)  ( x  2)( x  5) 2
f (4)  2
We say the image of 4 is 2
What is the image of 6
Sec 2.4: The Precise Definition of a Limit
f ( x)  2 x  1
What is the image of 4
9
7
What is the image of
the interval (3, 5)
5
3 4 5
Sec 2.4: The Precise Definition of a Limit
What is the image of the
interval (10/7, 10/3)
Sec 2.4: The Precise Definition of a Limit
f ( x)  ( x  2)( x  5) 2
f (4)  2
We say the image of 4 is 2
We say the pre-image of 2 is 4
What is the pre-image of -1
Sec 2.4: The Precise Definition of a Limit
f ( x)  2 x  1
10
7
What is the pre-image
of the interval (4, 10)
4
4
Sec 2.4: The Precise Definition of a Limit
What is the pre-image of the
interval (0.4, 0.6)
Sec 2.4: The Precise Definition of a Limit
What is the pre-image of the
epsilon-interval
f ( x)  0.1  0.06
Sec 2.4: The Precise Definition of a Limit
Definition:
lim f ( x)  L
x a
for every positive number

there is a positive number

such that
means that
if 0  x  a   then f ( x)  L   (*)
f (x)
L


For every epsilon-interval around L
there is a delta-interval around a
such that
The image of this delta-interval
contained inside the epsilon-interval


a
Sec 2.4: The Precise Definition of a Limit
Definition:
lim f ( x)  L
for every positive number

there is a positive number

such that
means that
x a
if 0  x  a   then f ( x)  L   (*)
f (x)
L




a
Sec 2.4: The Precise Definition of a Limit
Definition:
lim f ( x)  L
x a
for every positive number

there is a positive number

such that
means that
if 0  x  a   then f ( x)  L   (*)
f (x)
L


NOTE:
Give me any epsilon-interval, I
will give you a delta-interval
satisfying condition (*)
a
Sec 2.4: The Precise Definition of a Limit
Definition:
lim f ( x)  L
x a
for every positive number

there is a positive number

such that
means that
if 0  x  a   then f ( x)  L   (*)
Two types of problem
1) Given epsilon
find delta
graph
equation
2) Prove
Sec 2.4: The Precise Definition of a Limit
1) Given epsilon find delta (graph)
E1 – TERM121
Steps:
1) Identify a, L, f(x), epsilon
2) Draw horizontal lines y  L   , y  L  
3) Find the intersection points x1 , x2
4) Make sure that a  ( x1 , x2 )
5) Find the two deltas
1  a  x1 ,  2  a  x2
6) Now choose your delta to be
  min( 1 ,  2 )
Sec 2.4: The Precise Definition of a Limit
1) Given epsilon find delta (graph)
E1 – TERM121
Sec 2.4: The Precise Definition of a Limit
E1 – TERM102
Sec 2.4: The Precise Definition of a Limit
1) Given epsilon find delta (equation)
E1 – TERM122
Steps:
1) Identify a, L, f(x), epsilon
2) Solve these equations f ( x
1
)  L   , f ( x2 )  L  
3) Make sure that a  ( x1 , x2 )
a  ( x2 , x1 )
5) Find the two deltas
1  a  x1 ,  2  a  x2
6) Now choose your delta to be
  min( 1 ,  2 )
Sec 2.4: The Precise Definition of a Limit
Definition:
lim f ( x)  L
x a
for every positive number

there is a positive number

such that
means that
if 0  x  a   then f ( x)  L   (*)
Two types of problem
1) Given epsilon
find delta
graph
equation
2) Prove
Sec 2.4: The Precise Definition of a Limit
2) Prove
SOLUTION
3) Proof (showing
that this works)
2) guessing a value for
identify
f ( x)  4 x  5, a  3, L  7
Sec 2.4: The Precise Definition of a Limit
2) Prove
SOLUTION
1) Identify:
f ( x)  4 x  5, a  3, L  7
2) guessing a value for delta
if 0  x  3   then (4 x  5)  7  
(4 x  5)  7  
4 x 12  
This suggests that we should choose

x  3  4


4
3) Proof (showing that this works)
if 0  x  3   then (4 x  5)  7 
Thus
if 0  x  3   then (4 x  5)  7  
Therefore, by the definition of a limit
lim (4 x  5)  7
x 3
4 x  12  4 x  3  4 
 
 4  
4

Sec 2.4: The Precise Definition of a Limit
E1 – TERM111
Sec 2.4: The Precise Definition of a Limit
Sec 2.4: The Precise Definition of a Limit
Example: Use the graph of f ( x)  2 x  1
to find a number   0 such that
if x  4   then f ( x)  7  2
lim f ( x )  7
x4
if 0  x  a   then f ( x)  L  
f ( x)  2 x  1
9
5  f ( x)  9
7
3 x 5
5
choose   1
if x  4  1 then f ( x)  7  2
3 4 5
Sec 2.4: The Precise Definition of a Limit
f ( x)  ( x  2) 2  1
if x  4  0.2361 then f ( x)  3  1
 1
f ( x)  3  1


lim f ( x)  3
x4
0.2679
0.2361
  0.2361
x  4  0.2361
Sec 2.4: The Precise Definition of a Limit
Definition
:
lim f ( x)  L
x a
f (x)  5x  3
a1

for every positive number
means that
there is a positive number

such that
if 0  x  a   then f ( x)  L  
L 2
Part-1
Part-2
Part-3