Onesided limits
:
Suppose
f
(
x
) is a function defined for all values of
x
close
to but less than
a
; that is, suppose there exists
r
> 0 such
that
f
(
x
) is defined whenever
x
<
a
with 
x
–
a
 <
r
.
Then we
say that lim
x
→
a
–
f
(
x
) =
L
iff there for every
ε
> 0 there
exists
δ
> 0 such that for all
x
<
a
with 0 < 
x
–
a
 <
δ
we
have 
f
(
x
) –
L
 <
ε
.
Likewise:
Suppose
f
(
x
) is a function defined for all values of
x
close
to but greater than
a
; that is, suppose there exists
r
such that
f
(
x
) is defined whenever
x
>
a
with 
x
–
a
 <
r
.
Then we say
that lim
x
→
a
+
f
(
x
) =
L
iff there for every
ε
> 0 there exists
δ
> 0 such that for all
x
>
a
with 0 < 
x
–
a
 <
δ
we have

f
(
x
) –
L
 <
ε
.
Theorem: lim
x
→
a
f
(
x
) =
L
if and only if lim
x
→
a
–
f
(
x
) and
lim
x
→
a
+
f
(
x
) are both defined and both equal
L
.
Theorem: If
f
(
x
) and
g
(
x
) agree on some neighborhood of
a
(that is, if there exists
δ
0
such that
f
(
x
) =
g
(
x
) for all
x
satisfying 0 < 
x
–
a
 <
δ
0
) then lim
x
→
a
f
(
x
) = lim
x
→
a
g
(
x
).
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 Fall '11
 Staff
 Calculus, Logic, Limits, Equals sign, Mathematical terminology, L. Computing

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