10
(sin
x
)/
x
I call our world Flatland, not because we call it so, but
to make its nature clear to you, my happy readers, who
are privileged to live in Space.
—Edwin A. Abbott,
Flatland
(1884)
Students of calculus encounter the function
±
sin
x
²
/x
early in
their study, when it is shown that lim
x
→
0
±
sin
x
²
/x
=
1; this result
is then used to establish the differentiation formulas
±
sin
x
²
0
=
cos
x
and
±
cos
x
²
0
=
sin
x
. Once this has been done, however,
the function is soon forgotten, and the student rarely sees it
again. This is unfortunate, for this simplelooking function not
only has some remarkable properties, but it also shows up in
many applications, sometimes quite unexpectedly.
We note, to begin with, that the function is deﬁned for all
values of
x
except 0; but we also know that as
x
gets smaller and
smaller, the ratio
±
sin
x
²
/x
—provided
x
is measured in radians—
tends to 1. This provides us with a simple example of a
removable
singularity
: we can simply
deﬁne
the value of
±
sin 0
²
/
0tobe1
,
and this deﬁnition will assure the continuity of the function near
x
=
0.
Let us denote our function by
f
±
x
²
and plot it for various val
ues of
x
; the resulting graph is shown in ﬁgure 61. Two features
make this graph distinct from that of the function
g
±
x
²=
sin
x
:
ﬁrst, it is symmetric about the
y
axis; that is,
f
±
x
f
±
x
²
for all
values of
x
(in the language of algebra,
f
±
x
²
is an
even function
,
so called because the simplest functions with this property are of
the form
y
=
x
n
for even values of
n
). By contrast, the function
g
±
x
sin
x
has the property that
g
±
x
²=
g
±
x
²
for all
x
(func
tions with this property are called
odd functions
, for example
y
=
x
n
for odd values of
n
). To prove that
f
±
x
²=±
sin
x
²
/x
is even,
we simply note that
f
±
x
sin
±
x
²
/
±
x
²=±
sin
x
²
/
±
x
±
sin
x
²
/x
=
f
±
x
²
.
Second, unlike the graph of sin
x
, whose upanddown oscil
lations are conﬁned to the range from

1 to 1 (that is, the sine