as asymptote: values of the function become infinitely
large as x approaches the value 3 from the right, and
infinitely large and negative as x approaches the value
3 from the left.
A function y = f(x) has a horizontal asymptote y = L
if lim x→∞ f(x) = L or lim x→–∞ f(x) = L. For example,
the function
has horizontal asymptote y = 3, since
.
An asymptote need not be horizontal or vertical,
however. For example, the function
approaches the line y = x + 1 as x
becomes large, thus y = x + 1 is a “slant asymptote” for
the curve.
If a HYPERBOLA is given by the equation
,
then manipulating yields the equation
.
The right hand side tends to zero as x becomes large,
showing that the curve has slant asymptotes given by
, that is, by the lines
and
.
Extending the definition, we could say that the
curve
has the parabola y = x
2 as
an asymptote.
automaton (plural, automata) An abstract machine
used to analyze or model mathematical problems is
called an automaton. One simple example of an
automaton is a “number-base machine,” which consists
of a row of boxes extending infinitely to the left. One
places in this machine a finite number of pennies in the
rightmost box. The machine then redistributes the pennies according to a preset rule.
A “1 ← 2” machine, for example, replaces a pair
of pennies in one box with a single penny in the box
one place to the left. Thus, for instance, six pennies
placed into the 1 ← 2 machine “fire” four times to
yield a final distribution that can be read as “1 1 0.”
This result is the number six written as a BINARY NUMBER, and this machine converts all numbers to their
base-two representations.
A 1 ← 3 machine yields base-three representations,
and a 1 ← 10 machine yields the ordinary base-ten representations. The process of LONG DIVISION can be
explained with the aid of this machine.
Variations on this idea can lead to some interesting
mathematical studies. Consider, for example, a 2 ← 3
machine. This machine replaces three pennies in one
box with two pennies in the box one place to the left.
In some sense, this is a “base one and a half machine.”
For instance, placing 10 pennies in this machine yields
y
x
x
x
x
=
+ =
+
3
2
1
1
y
b
a
x
= −
y
b
a
x
=
b
a
y
x
−
=
2
2
0
b
a
y
x
b
x
−
=
2
2
2
2
x
a
y
b
2
2
2
2
1
−
=
x
x
= + +
+
2
1
1
1
y
x x
x
x
=
+
+ +
+
3
2
2
2
1
= + +
=
3
1 0 0
3
lim
lim
x
x
x
x x
x x
→∞
→∞
+ +
=
+ +
3
1
3
1
1 1
2
2
2
y
x
x
x
=
+ +
3
1
2
2
30 automaton
An automaton in action
large as x approaches the value 3 from the right, and
infinitely large and negative as x approaches the value
3 from the left.
A function y = f(x) has a horizontal asymptote y = L
if lim x→∞ f(x) = L or lim x→–∞ f(x) = L. For example,
the function
has horizontal asymptote y = 3, since
.
An asymptote need not be horizontal or vertical,
however. For example, the function
approaches the line y = x + 1 as x
becomes large, thus y = x + 1 is a “slant asymptote” for
the curve.
If a HYPERBOLA is given by the equation
,
then manipulating yields the equation
.
The right hand side tends to zero as x becomes large,
showing that the curve has slant asymptotes given by
, that is, by the lines
and
.
Extending the definition, we could say that the
curve
has the parabola y = x
2 as
an asymptote.
automaton (plural, automata) An abstract machine
used to analyze or model mathematical problems is
called an automaton. One simple example of an
automaton is a “number-base machine,” which consists
of a row of boxes extending infinitely to the left. One
places in this machine a finite number of pennies in the
rightmost box. The machine then redistributes the pennies according to a preset rule.
A “1 ← 2” machine, for example, replaces a pair
of pennies in one box with a single penny in the box
one place to the left. Thus, for instance, six pennies
placed into the 1 ← 2 machine “fire” four times to
yield a final distribution that can be read as “1 1 0.”
This result is the number six written as a BINARY NUMBER, and this machine converts all numbers to their
base-two representations.
A 1 ← 3 machine yields base-three representations,
and a 1 ← 10 machine yields the ordinary base-ten representations. The process of LONG DIVISION can be
explained with the aid of this machine.
Variations on this idea can lead to some interesting
mathematical studies. Consider, for example, a 2 ← 3
machine. This machine replaces three pennies in one
box with two pennies in the box one place to the left.
In some sense, this is a “base one and a half machine.”
For instance, placing 10 pennies in this machine yields
y
x
x
x
x
=
+ =
+
3
2
1
1
y
b
a
x
= −
y
b
a
x
=
b
a
y
x
−
=
2
2
0
b
a
y
x
b
x
−
=
2
2
2
2
x
a
y
b
2
2
2
2
1
−
=
x
x
= + +
+
2
1
1
1
y
x x
x
x
=
+
+ +
+
3
2
2
2
1
= + +
=
3
1 0 0
3
lim
lim
x
x
x
x x
x x
→∞
→∞
+ +
=
+ +
3
1
3
1
1 1
2
2
2
y
x
x
x
=
+ +
3
1
2
2
30 automaton
An automaton in action