How do you find asymptotic discontinuity?
Answers
If a function
f
(
x
)
has a vertical asymptote at
a
, then it has a asymptotic (infinite) discontinuity at
a
. In order to find asymptotic discontinuities, you would look for vertical asymptotes. Let us look at the following example.
f
(
x
)
=
x
+
1
(
x
+
1
)
(
x
−
2
)
In order to have a vertical asymptote, the function has to display "blowing up" or "blowing down" behaviors. In the case of a rational function like
f
(
x
)
here, it display such behaviors when the denominator becomes zero.
By setting the denominator equal to zero,
(
x
+
1
)
(
x
−
2
)
=
0
⇒
x
=
−
1
,
2
Now, we have a couple of candidates to consider. Let us make sure that there is a vertical asymptote there.
Is
x
=
−
1
a vertical asymptote?
lim
x
→
−
1
(
x
+
1
)
(
x
+
1
)
(
x
−
2
)
by cancelling out
(
x
+
1
)
's,
=
lim
x
→
−
1
1
x
−
2
=
1
1
−
2
=
−
1
≠
±
∞
,
which means that
x
=
−
1
is NOT a vertical asymptote.
Is
x
=
2
a vertical asymptote?
lim
x
→
2
+
x
+
1
(
x
+
1
)
(
x
−
2
)
by cancelling out
(
x
+
1
)
's,
=
lim
x
→
2
+
1
x
−
2
=
1
0
+
=
+
∞
,
which means that
x
=
2
IS a vertical asymptote.
Hence,
f
has an asymptotic discontinuity at
x
=
2
.
I hope that this was helpful.
If the left or right side limits at x = a are infinite or do not exist, then at x = a there is an essential discontinuity or infinite discontinuity. At x = 2 there is an essential discontinuity because there is no right side limit.