Math 165, Section D
Professor Lieberman
December 6, 2002
QUIZ #14
Directions:
(1) Include this information with your answer: Your name, the
course number (165), your section number (D11, D12, D13,
D14, D15, or D16), and the quiz number.
(2) Starting Friday, December 6, there will be a box labeled “ungraded Math 165 Quizzes” outside my office (422 Carver). Place
your answers in this box. DO NOT give your answers to your
TA. No earlier quizzes will be graded
(3) This quiz is due at 4 p.m., Monday, December 9.
Question: Evaluate the limit
x2
1
lim 1 +
.
x→∞
x
Answer: You must use L’Hospital’s Rule. First
x2
1
lim 1 +
= lim ef (x)
x→∞
x→∞
x
with
1
2
f (x) = x ln 1 +
,
x
and
ln(1 + 1/x)
lim f (x) = lim
.
x→∞
x→∞
x−2
This is the indeterminate form 00 , so we can apply L’Hospital’s rule:
ln(1 + 1/x)
[1/(1 + 1/x)][−x−2 ]
x2
=
lim
=
lim
= ∞.
x→∞
x→∞
x→∞ 2(1 + x)
x−2
−2x−3
Therefore,
x2
1
lim 1 +
= ∞.
x→∞
x
lim