lim
x
a
f
x
L

P13
1.3
THE LIMIT OF A FUNCTION
Roughly speaking, this says that the values of
f
(
x
) tend to get closer and closer to the number
L
as
x
gets closer and closer to the number
a
(from
either side of
a
) but
x
a
.
An alternative notation for
is
as
which is usually read “
f
(
x
) approaches
L
as
x
approaches
a
.”
lim
x
a
f
x
L
( )
f x
L
x
a

P14
1.3
THE LIMIT OF A FUNCTION
Notice the phrase “but
x
a
” in the definition
of limit.
This 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
.

P15
1.3
THE LIMIT OF A FUNCTION
Figure 2 shows the graphs of three functions.
Note that, in the third graph,
f
(
a
) is not defined and,
in the second graph,
.
However, in each case, regardless of what happens
at
a
, it is true that
( )
f a
L
lim
( )
.
x
a
f x
L

P16
1.3
Example 2
Guess the value of
.
SOLUTION
Notice that the function
f
(
x
) = (
x
– 1)/(
x
2
– 1) is not
defined when
x
= 1.
However, that doesn’t matter—because the
definition of
says that we consider values of
x
that are close to
a
but not equal to
a
.
2
1
1
lim
1
x
x
x
lim
( )
x
a
f x

P17
1.3
Example 2 SOLUTION
The tables give values of
f
(
x
) (correct to six decimal
places) for values of
x
that
approach 1 (but are not
equal to 1).
On the basis of the values, we
make the guess that
2
1
1
lim
0.5
1
x
x
x

P18
1.3
THE LIMIT OF A FUNCTION
Example 2 is illustrated by the graph of
f
in
Figure 3.

P19
1.3
THE LIMIT OF A FUNCTION
Now, let’s change
f
slightly by giving it the
value 2 when
x
= 1 and calling the resulting
function
g
:
This new function
g
still
has the same limit as
x
approaches 1.
See Figure 4.
2
1
if
1
1
2
if
1
x
x
g x
x
x

P20
1.3
Example 3
Estimate the value of
.
SOLUTION
The table lists values of the function for several
values of
t
near 0.
As
t
approaches 0,
the values of the function
seem to approach
0.16666666…
So, we guess that:
2
2
0
9
3
lim
t
t
t
2
2
0
9
3
1
lim
6
t
t
t

P21
1.3
THE LIMIT OF A FUNCTION
What would have happened if we had taken
even smaller values of
t
?
The table shows the results from one calculator.
You can see that something strange seems to be
happening.
If you try these
calculations on your own
calculator, you might get
different values but,
eventually, you will get
the value 0 if you make
t
sufficiently small.

P22
1.3
THE LIMIT OF A FUNCTION
Does this mean that the answer is really 0
instead of 1/6?
No, the value of the limit is 1/6, as we will show in
the next section.
The problem is that the calculator gave false
values because
is very close to 3 when
t
is
small.

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- Fall '19
- Calculus, Limit, One-sided limit