This preview shows pages 1–2. Sign up to view the full content.
Department of Mathematics, University of Toronto
Term Test 1 – November 15, 2006
MAT 137Y, Calculus!
Time Alloted: 1 hour 50 minutes
1.
Evaluate the following limits. Do not use L’Hˆopital’s Rule to evaluate the limit.
(7%)
(i)
lim
x
→
0
(
x

1
)
2

1
x
2
+
6
x
.
(7%)
(ii)
lim
t
→
0
sin
2
(
5
t
)
3
t
2
.
(7%)
(iii)
lim
x
→
4
+
(
4

x
)

3
x

14


4

x

.
(7%)
(iii)
lim
x
→
0
3

√
9

x
2
x
2
.
2.
(7%)
(i)
Solve the inequality
x
2

3
x
x
4

1
≤
0. Express your answer as a union of intervals.
(ii)
Suppose sin
x
=
3
4
and
π
2
≤
x
≤
π
. Find the exact value of each of the following expres
sions.
(6%)
(a)
tan
x
.
(4%)
(b)
cos2
x
.
3.
(5%)
(a)
Give the precise
ε
,
δ
deﬁnition of the following statement: lim
x
→
a
f
(
x
) =
L
.
(12%)
(b)
Prove that lim
x
→
3
x
2
+
1
1

x
=

5 directly using the precise deﬁnition of limit.
4.
Consider the sequence of numbers
x
1
=
√
1
,
x
2
=
q
1
+
√
1
,
x
3
=
r
1
+
q
1
+
√
1
,
x
4
=
s
1
+
r
1
+
q
1
+
√
1
,...,
so
x
n
containes
n
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 UPPAL
 Calculus, Limits

Click to edit the document details