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irrational number
r
such that
r
sqrt(2)
is rational?
Hint: The answer is in Wednesday’s lecture (sort of!).
Solution: We can easily prove that either
r
= sqrt(2) or
r
=
sqrt(2)
sqrt(2)
works, but the proof won’t tell us which!
Proof: If sqrt(2)
sqrt(2)
is rational, then
r
= sqrt(2) works.
If
sqrt(2)
sqrt(2)
is irrational, then
r
= sqrt(2)
sqrt(2)
works, because
then
r
sqrt(2)
= (sqrt(2)
sqrt(2)
)
sqrt(2)
= sqrt(2)
sqrt(2)sqrt(2)
= sqrt(2)
2
= 2.
As it happens, it’s been proved that sqrt(2)
sqrt(2)
is irrational.
Last time we found empirically (i.e. with a calculator) that
lim
x
→
0+
(1+
x
)
1/
x
and lim
x
→
0–
(1+
x
)
1/
x
both are about 2.7.
In
fact, both limits are equal to the famous irrational number
e
= 2.718…
That is:
e
= lim
x
→
0
(1+
x
)
1/
x
(a twosided limit).
We can also write
e
as lim
n
→∞
(1+1/
n
)
n
and lim
n
→∞
(1–1/
n
)
–
n
(Why?
..?.
.
Put
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 Fall '11
 Staff
 Calculus

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