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Study Sheet for Test 1
Honors Calculus
Keesling
Test on 9/24/08
1.
Compute the following limits.
(a)
lim
x
!
2
x
3
"
2
x
( )
(b)
lim
x
!
0
x
"
sin
1
x
( )
(c)
lim
x
!
1
x
3
"
1
x
2
"
1
(d)
lim
x
!
0
x
+
1
"
1
x
(e)
lim
x
!
0
tan(
x
)
sin(
x
)
(d)
lim
x
!
0
x
"
cos
1
x
2
( )
2.
Determine which numbers are represented by the following decimal expansions
(a)
.99
9
(b)
203.71914
(c)
!
1.2031
3.
Define a rational number.
Give an argument that the following decimal
expansion cannot represent a rational number
.1010010001
!
100
!
0
n
zeros
"#$
1
!
.
4.
Prove that the following limit holds for any
x
<
1
x
n
n
=
0
!
"
=
1
1
#
x
.
Also, be able to
use this formula:
(a)
1
3
( )
n
n
=
0
!
"
(b)
!
1
9
( )
n
n
=
1
"
#
(c)
1
3
( )
n
n
=
2
!
"
(d)
3
n
n
=
0
3000
!
5.
Newton used the tangent line to approximate a zero of a function
f
.
He did this
by taking a value where
f
is near zero, say
x
0
.
The point
x
0
is the first
approximation of a zero of
f
.
He then considered where the tangent line through
the point
(
x
0
,
f
(
x
0
))
crossed the
x
–axis. Call this point
x
1
.
This should be a
better approximation the zero of
f
than
x
0
.
The process continues.
Demonstrate
the formula for
x
n
+
1
in terms of
x
n
given below.
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 Spring '07
 JURY
 Calculus, Limits

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