Mathematical Definition of Limit (    Definition)
 (Epsilon) ,
I.
Recall First Lesson:
(Delta)
lim f ( x)  L means that
xa
" f ( x) is near to L if x is near to a".
"| f ( x) - L | is near to 0 if | x - a | is near to 0"
Comment: The term “Near” is not clear. One can ask “How much near?”
The Mathematical Answer to this Question is as follows:
lim f ( x)  L ……… (1)
xa
means that
For a given Positive Number  [  can be very samll],
we can find a Positive Number  [may be very small] so that
| f ( x)  L |  when 0< |x  a | <  .....(2)
Terminology: f ( x) is  -near to L when x is  -near to a.
Another way to write (2)
L    f ( x)  L   when a   x  a   ; x  a .....(3)
Graphical Concept of Limit
f (x)
L+ 
L 
L
Choice of
minimum
( 1, 2
)
a-
a
1
a+
2
Note: 1.The Graph of f (x) ; x a ; lies in the Box bounded by the lines
y=L-  , y=L+  , x=a- , x=a+ …………..(4)
2. The Mid Points of Length and width of the Box are at a and L respectively.
3. is the Largest Number which satisfies (4)
II.   N Definition for lim f ( x)  L
x 
For a given Positive Number  [  can be very samll],
we can a Positive Number N [may be very Large] so that
| f ( x)  L |  when x > N
f (x)
L+ 
L
L 
N
Graphical Concept: The Graph of f (x) lies between two Horizontal lines
y=L-  , y=L+  , when x > N
III. N   Definition for lim f ( x)  
xa
For a given Positive Number N [ may be very Large],
we can a Positive Number  [may be very small] so that
f ( x)  N when 0<|x  a | 
IV. N   Definition for
lim f ( x)  
x a
For a given Negative Number N ,
we can find a Positive Number  so that
N
f ( x)  N when |x  a | 
0
a-
a
a-
a
a+
a+
Graphical Concept
N
The Graph of f (x) lies above the
Horizontal line y=N
when a -
<x< a+ ;x a
Questions(2.3)1(a).
Find the Largest Open Interval
centered at 4
on the x-axis
such that
Questions (2.3) 39
Using   N Definition of limit, find the
value of Positive N for the following:
x
Questions(2.3)1(b).
Find the Largest Open Interval
centered at the Origin
on the x-axis
such that
for each x in the interval
“the value of the function f (x) = x2 ”
is written within
0.01 unit of the number f (4)=16.
Hint: Find L, a ,
. Use the Hint of Ex. 1(a).
Questions (2.3) 4 (a)
Find the value of x1 and x2 in the figure:
1+0.1
Questions (2.3) 43
Using   N Definition of limit, find the
value of Negative N for the following:
4 x 1
lim
2 ;
2
x

5
x 
  0.1
Hint
4 x 1
 2  0.1
2 x 5
Change it to
x<N
f (x) =1/x
Questions (2.3) 55
1
Using N   Definition of limit ,
1 – 0.1
prove that
x1 1 x2
Hint
Note that f (x1) = 1+ 0.1.
Solve the Equation for x1.
Questions (2.3) 16
Using    Definition of limit, find the
value of
for the following:
lim x  3
x 9
  0.001
Hint
Note that
x 3  0.001. Change it to
x 9 
lim
x3
1
 x 3  2

Hint
Choose any Positive number N
Consider
f (x) > N.
Change it to
x 3  
Find the value of