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18.01 Single Variable Calculus
Fall 2006
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CONTINUITY
AND
DISCONTINUITY
1. Onesided
limits
We
begin
by
expanding the notion of limit to include what are called
onesided limits,
where
z
approaches a only from one side

the right or the left. The terminology and
notation is:.
righthand limit
lim
f(x)
(z
comes from the right, x
>
a)
lefthand limit
lim
f(z)
(x comes from the left,
z
<
a)
X
ML_
Since we use limits informally, a few examples will be enough to indicate the usefulness of
this idea.
1/x
2
.1
1
Ex. 1
Ex.2
Ex. 3
Ex.4
Example
1.
lim
V/l
2
= 0
=1
=
*1+
(As the picture shows, at the two endpoints of the domain, we only have a onesided limit.)
z
0
Example
2.
Set
f(z)=
1
X 0.
Then
>
,*+
lim
f()
=
1,
X00+
lim
f()
=1.
1
1
Example
3.
li

=
oo,
lim

=
oo
_O0+ X
20
2
1
1
Example 4.
li n
=oo,
lim

oo
.
4
0+
2
2
2o z
The relationship between the onesided limits and the usual (twosided) limit.is given
by
(1)
.
IC=+&
lim
f(x)
=
L
lim
f(x)
=
L
and lim
f()
=
L
240+
In words, the (twosided) limit exists if and only if both onesided limits exist and are equal.
This shows for example that in Examples
2
and
3
above,
lim
f(z)
does not exist.
Students often say carelessly that
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 Spring '11
 Blake
 Math, Continuity, Limits

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