MAT 319/320
Correction of HW4
Exercise 1.
Page 67, #6a.
Proof.
By the sum rule, the limit of
(2+1/
n
)
is equal to
2
. By the product rule, the limit of
(2+1/
n
)
2
is 2.2
=4
.
square
Exercise 2.
Page 67, #9.
Proof.
One has
y
n
=
n
+1
√

n
√
=
1
n
+1
√
+
n
√
(multiply the numerator and denominator by
the conjugate quantity).
But now one has
0
lessorequalslant
y
n
lessorequalslant
1
n
√
. Therefore if one proves that
(1/
n
√
)
converges to zero, the
squeeze theorem implies that
(
y
n
)
converges itself to zero.
Fix any
ε >
0
, then by the archimedean property there exists a natural number
K
such that
K > ε
2
, but this implies that for any
n
greaterorequalslant
K
one has
n > ε
2
, implying
1
n
√
< ε
, thus we proved
that
(1/
n
√
)
converges to zero, and hence
(
y
n
)
converges to zero.
Now
n
√
y
n
=
n
√
n
+1
√
+
n
√
=
n
√
n
√
.
1
(1+1/
n
)
+1
=
1
1+
1+
1
n
. By the square root theorem
1+
1
n
radicalBig
converges to
1
. By the quotient theorem (which applies because the limit of the denominator is
nonzero), one knows that
n
√
.y
n
converges to
1/2
.
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 Fall '10
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 Limit, Xn

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