Test 1  Solutions
Math 102
Problem 1:
(
5 points
) Consider the function
f
(
x
) =
1
+
1
1
+
1
x
.
Compute the following limits:
lim
x
→
0
f
(
x
)
,
lim
x
→
1

f
(
x
)
,
lim
x
→
∞
f
(
x
)
.
Solution:
The easiest way to compute the limits is to first rewrite the function:
f
(
x
) =
1
+
1
1
+
1
x
=
1
+
1
x
+
1
x
=
1
+
x
x
+
1
=
2
x
+
1
x
+
1
Then
lim
x
→
0
f
(
x
) =
lim
x
→
0
2
x
+
1
x
+
1
=
1;
lim
x
→
1

f
(
x
) =
lim
x
→
1

2
x
+
1
x
+
1
= +
∞
;
lim
x
→
∞
f
(
x
) =
lim
x
→
∞
2
x
+
1
x
+
1
=
2;
1
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Problem 2:
(
5 points
) True or False?
T
F
No rational function can have two horizontal asymptotes.
T
F
No function can have more than two horizontal asymptotes.
T
F
There exist differentiable functions which are not continuous.
T
F
If
f
0
(
3
) =
g
0
(
3
)
then the tangent lines to the graphs of
f
and
g
at
x
=
3 coincide.
T
F
There exists a function which is continuous on
[
0
,
1
]
and has a vertical asymptote at
x
=
1
2
.
Solution:
The answers are as follows:
True:
We know a useful rule telling us how to decide whether a rational function has horizontal
asymptotes. According to that rule, there is either no asymptote (when the top has degree larger than
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 Spring '10
 MARYAMNAMAZI
 Calculus, Logic, Derivative, Limits, lim, Limit of a function, lim g

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