WOLFRAM|DEMONSTRATIONS PROJECT

Squeeze Theorem

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n
0
1
2
3
zoom
Let
f
,
g
, and
h
be functions satisfying
f(x)≤g(x)≤h(x)
for all
x
near
a
, except possibly at
a
. By the squeeze theorem, if
lim
xa
f(x)=
lim
xa
h(x)=L
then
lim
xa
g(x)=L
. Hence,
lim
x0
n
x
sin
1
x
equals zero if
n=1,2
, or
3
, since
n
x
sin
1
x
is squeezed between
-|
n
x
|
and
|
n
x
|
. The theorem does not apply if
n=0
, since
sin
1
x
is trapped but not squeezed. For
n=0
the limit does not exist, because no matter how close
x
gets to zero, there are values of
x
near zero for which
sin
1
x
=1
and some for which
sin
1
x
=-1
.