A Primer on Limits
1. Perhaps the simplest kind of limit is a sequential limit: a sequence b : N R has a limit L provided each neighborhood
of L contains a tail of b.
The two terms neighborhood and tail need dening:
(a) A neighborhood of a real number is
Worked Examples
Example 1: Evaluate the limit: lim e1/x ln(1 + x). The indeterminant form here is 0.
x0+
(a) One approach:
(i) Change the indeterminant form to /: e1/x ln(1 + x) =
e1/x
.
[ln(1 + x)]1
(ii) Apply LHospitals rule:
lim
x0+
Since
e1/x
e1/x (1/
Math 1720
Problem 1
Exam 4 Solutions
Compute
C
F dr for each vector eld F and curve C below:
(a). F(x, y ) = (y, x) and C is the unit circle (in the plane) directed counterclockwise.
A Solution: r(t) = (cos t, sin t) for 0 t 2 parameterizes the unit circl
Math 1720
Exam 3 Solutions
Problem 1
(a). Find the work done by a constant force F = (1, 1, 4) N on an object as it moves along a the straight line segment
from A = (0, 0, 0) to B = (1, 1, 1).
A Solution:
Work done by F( in Joules) = F (B A) = (1, 1, 4) (
Math 1720
Exam 2 Solutions
Problem 1
(a).
x cos x dx
Integrate by parts:
R
x cos x dx = x sin x
R
Z
A Solution:
e
e
sin x dx = x sin x + cos x + C .
e
Z
(b).
ln t dt
1
A Solution:
e
1
Z
dt = (e ln e ln 1) (e 1) = 1.
ln t dt = t ln t
Integrate by parts:
1
Math 1720
Exam 1 Solutions
Feb 23, 2009
Problem 1 Compute
(a) lim t ln t
+
t0
A Solution:
lim t ln t = lim
t0+
t0+
1/t
ln t
= lim
= lim t = 0 (LHospitals rule was used at the second equality).
+ 1/t2
1/t
t 0
t0+
(b) lim [ln(x + 2) ln x]
x
A Solution:
lim