Prove: lim 4 = 4
x ->2
Answers
Step-by-step explanation:
Given a function
f
(
x
)
, we say that the limit as
x
approaches
a
of
f
(
x
)
is
L
, denoted
lim
x
→
a
f
(
x
)
=
L
, if for every
ε
>
0
there exists a
δ
>
0
such that
0
<
|
x
−
a
|
<
δ
implies that
|
f
(
x
)
−
L
|
<
ε
.
In more intuitive terms, we say that
lim
x
→
a
f
(
x
)
=
L
if we can make
f
(
x
)
arbitrarily "close" to
L
by making
x
close enough to
a
.
Now, to use this in a proof with
f
(
x
)
=
x
2
,
a
=
2
, and
L
=
4
:
Proof: Let
ε
>
0
be arbitrary. Let
δ
=
√
ε
+
4
−
2
(note that
δ
>
0
as
√
ε
+
4
>
√
4
=
2
).
Suppose
|
x
−
2
|
<
δ
. Then
−
δ
<
x
−
2
<
δ
⇒
−
δ
+
4
<
x
+
2
<
δ
+
4
⇒
−
δ
−
4
<
x
+
2
<
δ
+
4
⇒
|
x
+
2
|
<
δ
+
4
With that, then if
0
<
|
x
−
2
|
<
δ
, we have
∣
∣
x
2
−
4
∣
∣
=
|
x
−
2
|
⋅
|
x
+
2
|
<
δ
(
δ
+
4
)
=
(
√
ε
+
4
−
2
)
(
√
ε
+
4
−
2
+
4
)
=
(
√
ε
+
4
)
2
−
2
2
=
ε
We have shown that for any
ε
>
0
there exists a
δ
>
0
such that
0
<
|
x
−
2
|
<
δ
implies
∣
∣
x
2
−
4
∣
∣
<
ε
. Thus, by the
ε
−
δ
definition of a limit,
lim
x
→
2
x
2
=
4