國立彰化師範大學
110學年度碩士班招生考試試題
系所：
數學系
(選考丙)、
科目：微積分
統計資訊研究所
(選考丙)
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以下皆是計算題與證明題，必須書寫計算或證明過程。
1. (10%) Prove, by the
definition of limits,
lim
x
a
x
a
(
0
a
).
2. (20%) Evaluate the following limits:
(a)
2
0
lim
1 cos
x
x
x
(b)
5
ln
lim
x
x
x
(c)
2
1/
0
lim(cos )
x
x
x
(d)
lim(
)
x
x
x
3. (20%) Find the following integrals:
(a)
x
e
dx
(b)
3
2
1
2
4
2
5
x
dx
x
x
4. (30%) Evaluate the following double integrals:
(a)
D
xy dxdy
, where
D
is the region bounded by the line
1
y
x
and the parabola
2
2
6
y
x
.
(b)
2
(
)
R
x
y dxdy
, where
R
is the region lying between the two circles
2
2
1
x
y
and
2
2
5.
x
y
(c)
2
2
(
2
5
)
x
xy
y
e
dxdy
5. (20%) Test the series below for convergence.
(a)
2
0
2
5
1
n
n
n
n
(b)
1
1
ln(
2)
ln
2
tan ( )
n
n
n
n
(c)
99
1
2
n
n
n
(d)
2
1
ln
n
n
n