Chapter 2: Limits
2.2 Definitions of Limits
Definition 1: Suppose the function f is defined for all x is near a except possibly at a . If
f ( x ) is arbitrary close to L (as close to L as we like) for all x sufficiently close (but not
equal) to a , we write
lim
f (x)  L
x a
and say the limit of f ( x ) as x approaches a equals L .
An alternative notation for
lim
f (x)  L
x a
f (x)  L
is
as x  a
which is usually read “ f ( x ) approaches L as x approaches a .”
Remark 1:
a) We say that lim f ( x )  L if f ( x ) gets closer and closer to L as x gets closer to a
x a
from both sides of a .
b) The value of lim f ( x ) (if it exists) depends upon the values of f near a , but it does
x a
not depend on the value of f ( a ) .
Finding Limits from a Graph:
Use the given graph to find the value of the limit by choosing values for
sufficiently close to a (on either side of a ) but not equal to a .
Example 1: Use the graph of
a)
f
x
that are
to determine the following values, if possible:
f (1 )
b) lim f ( x )
x1
c)
f (2)
d) lim f ( x )
x 2
e)
f (3)
f) lim f ( x )
x 3
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Mr. Ahmad Al Share’
Chapter 2: Limits
2.2 Definitions of Limits
Remark 2: The phrase “but x  a ” in the definition of limit means that in finding the limit
of f ( x ) as x approaches a , we never consider x  a . In fact, f ( x ) need not even be
defined when x  a . The only thing that matters is how f is defined near a .
The following figures show the graphs of three functions. Note that in part (c), f ( a ) is not
defined and in part (b), f ( a )  L . But in each case, regardless of what happens at a , it is
true that lim f ( x )  L .
x a
Finding Limits from a Table:
Form suitable table by taking values for x that are sufficiently close to a (on either side of a
) but not equal to a .
Example 2: Create a table of values of f ( x ) 
x 1
x 1
corresponding to value of x near 1 .
Then make a conjecture about the value of lim f ( x ) ?
t1
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Mr. Ahmad Al Share’
Chapter 2: Limits
2.2 Definitions of Limits
One – Side Limits:
Definition 2:
a) Right – hand limit: Suppose that f is defined for all x near a with x  a , if f ( x ) is
arbitrary close to L for all x sufficiently close to a with x  a , we write
lim
x a
f (x)  L

and say the limit of f ( x ) as x approaches a from the right equal L .
b) Left – hand limit: Suppose that f is defined for all x near a with x  a , if f ( x ) is
arbitrary close to L for all x sufficiently close to a with x  a , we write
lim
x a
f (x)  L

and say the limit of f ( x ) as x approaches a from the left equal L .
x 8
3
Example 3: Let f ( x ) 
4(x  2)
. Use table and graphs to make a conjecture about the
values of lim f ( x ) and lim f ( x ) ?
x 2
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
x 2

Mr. Ahmad Al Share’
Chapter 2: Limits
2.2 Definitions of Limits
By comparing the definition with the definitions of one-sided limits, we see that the following
is true.
Theorem 1: Relationship between One – Sided and Two – Sided Limits
Assume f is defined for all x near a except possibly at a . Then lim f ( x )  L if and
x a
only if lim f ( x )  L and lim f ( x )  L .
x a

x a

Example 4: The function H is defined by
0
H (t )  
1
if
t  0
if
t  0
Find lim H ( t ) and lim H ( t ) ?
x 0

x 0

Example 5: Use the graph of the function g to state the values (if they exist) of the
following:
a) lim g ( x )
x 2

b) lim g ( x )
x 2

c) lim g ( x )
x 2
d) g ( 2 )
e) lim g ( x )
x 5

f) lim g ( x )
x 5

g) lim g ( x )
x 5
h) g ( 5 )
 
:
 x 
Example 6: Use the following graph to investigate lim sin 
x 0
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Mr. Ahmad Al Share’