University of Engineering and Technology Lahore
Department of Electrical Engineering
Midterm Practice Problems
Calculus I
January 5, 2010
Total marks: 30
Time 90 mins
Instructions
•
Give
brief
and
concise
answers. Do not needlessly fill pages.
1. Compute lim
x
→∞
1
x
+
1
ln
x
using the Sandwich theorem and other limit the
orems.
2. Let
F
(
x
) =
x
2

1

x

1

. Find
(a) lim
x
→
1
+
F
(
x
)
(b) lim
x
→
1

F
(
x
).
Does the limit lim
x
→
1
F
(
x
) exist?
3. If lim
x
→
a
[
f
+
g
] (
x
) = 2 and lim
x
→
a
[
f

g
] (
x
) = 1, what is lim
x
→
a
[
fg
] (
x
)?
4. Show that when
p >
0,
lim
x
→∞
ln
x
x
p
= 0
5. Find a point on the parabola
y
= 2
x
2
that is closest to the point (1
,
4) .
6. Consider the function
w
defined by
w
(
x
) =
f
(
g
(
h
(
x
))) +
k
(
x
)
where
f
,
g
,
h
and
k
are functions which are differentiable everywhere on
the real line. Compute
w
0
(
x
).
7. Compute the limit (Hint: Write
d
dx
e
x
as a limit)
lim
x
→
1
e
x

e
x
2

1
8. Construct a function
f
, defined on all of
R
with the following property:
For each
M >
0, there exists a
δ >
0 such that
f
(
x
)
≥
M
for all
x
with
0
<

x

< δ
. Show that this property implies that lim
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 '10
 RazaSuleman
 Electrical Engineering, Calculus, Limit, Continuous function, University of Engineering and Technology Lahore Department of Electrical Engineering

Click to edit the document details