MA140-01
3/25/09
Page 1
Major Quiz 4 SOLUTIONS
No Calculator Portion
Somebody’s gonna lotto, may as well be ________________________
Show all your work and explain your answers completely. I cannot give partial credit for answers that are both wrong and
unexplained. Even correct "bottom line" answers that are mysterious and unsupported will not be considered completely correct.
Show me what you are thinking. Try to keep your answers neat and organized so that I can follow them easily.
1.) Find each of the following limits, if they exist.
(a)
3
5
0
1
sin
6
lim
x
x x x
x
→
− +
0
0
(b)
(
)
2
1 ln
0
lim x
xx
+
+
→
Apply L’hopital’s rule.
2
1 ln
0 0
lim ln lim ln
x
x x
y x
+ +
+
→ →
=
2
4
0
1
cos 1
0
2
lim
5 0
x
x x
x
→
− +
2
1 ln
0 0
lim ln lim ln
x
x x
y x
+ +
+
→ →
=
Apply L’hopital’s rule.
( )
0 0
2
lim ln lim ln
1 ln
x x
y x
x
+ +
→ →
=
+
3
0
sin 0
lim
20 0
x
x x
x
→
− +
0 0
2ln
lim ln lim
1 ln
x x
x
yx
+ +
→ →
∞
=
+ ∞
Apply L’hopital’s rule.
'
0 0
2
lim ln lim
1
L H
x x
x
y
x
+ +
→ →
∞
=
∞
2
0
cos 1 0
lim
60 0
x
x
x
→
− +
(
)
0 0
lim ln lim 2
x x
y
+ +
→ →
=
Apply L’hopital’s rule....again.
(
)
ln 2
0 0
lim ln lim y
x x
y e e
+ +
→ →
= =
0
sin 0
lim
120 0
x
x
x
→
Apply L’hopital’s rule....one more time.
0
cos 1
lim
120 120
x
x
→→
