The function f(x)
x² – 9
is discontinuous at
x - 3
Answers
Answer:
In order for a function
f
(
x
)
to be continuous at a given
x
-value
a
, the following condition must be satisfied:
[
1
]
lim
x
→
a
f
(
x
)
=
f
(
a
)
What this is saying is that, as
x
gets closer to
a
,
f
(
x
)
should also get closer to
f
(
a
)
.
For the given function
f
(
x
)
, the limit on the left-hand side of
[
1
]
will evaluate correctly. You'll end up with
lim
x
→
3
x
2
−
9
x
−
3
=
6
.
However, the right-hand side of
[
1
]
presents a problem: what is
f
(
x
)
when
x
=
3
?
The answer is, it is not defined, because at that point, we have
f
(
x
)
"equal" to
0
0
:
f
(
3
)
=
3
2
−
9
3
−
3
=
9
−
9
0
=
0
0
And this "value" of
0
0
is indeterminate.
Thus, the function "breaks" at
x
=
3
, and so, because there is no
f
(
3
)
, it is not possible to say
lim
x
→
3
f
(
x
)
=
f
(
3
)
.
Thus,
f
(
x
)
is not continuous at
x
=
3
.