What is rational function and how do you find domain, vertical and horizontal asymptotes. Also what is "holes" with all limits and continuity and discontinuity?
Answers
A rational function is where there are
x
's under the fraction bar.
The part under the bar is called the denominator .
This puts limits on the domain of
x
, as the denominator may not work out to be
0
Simple example:
y
=
1
x
domain :
x
≠
0
This also defines the vertical asymptote
x
=
0
, because you can make
x
as close to
0
as you want, but never reach it.
It makes a difference whether you move toward the
0
from the positive side of from the negative (see graph).
We say
lim
x
→
0
+
y
=
∞
and
lim
x
→
0
−
y
=
−
∞
So there is a discontinuity
graph{1/x [-16.02, 16.01, -8.01, 8.01]}
On the other hand: If we make
x
larger and larger then
y
will get smaller and smaller, but never reach
0
. This is the horizontal asymptote
y
=
0
We say
lim
x
→
+
∞
y
=
0
and
lim
x
→
−
∞
y
=
0
Of course ratinal functions are usually more complicated, like:
y
=
2
x
−
5
x
+
4
or
y
=
x
2
x
2
−
1
but the idea is the same
In the latter example there are even two vertical asymptotes, as
x
2
−
1
=
(
x
−
1
)
(
x
+
1
)
→
x
≠
+
1
and
x
≠
−
1
graph{x^2/(x^2-1) [-22.8, 22.81, -11.4, 11.42]}