15
True or False. In the following exercises, justify your answer with a proof or a counterexample.
1. A function has to be continuous at if the exists.
2. You can use the quotient rule to evaluate .
Solution
False
3. If there is a vertical asymptote at for the function , then is undefined at the point .
4. If does not exist, then is undefined at the point .
Solution
False. A removable discontinuity is possible.
5. Using the graph of , find each of the following or explain why it does not exist.
In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.
6.
Solution
5
7.
8.
Solution
9.
10.
Solution
DNE
11.
12.
Solution
13.
14.
Solution
−4
15.
16.
Solution
17.
18.
Solution
2
19.
20.
Solution
21.
22.
Solution
23.
24.
Solution
25.
In the following exercises, evaluate the limits to infinity.
26.
Solution
27.
28.
Solution
29.
30.
Solution
31.
In the following exercises, use the squeeze theorem to prove the limit.
32.
Solution
Since , then . Since , it follows that .
33.
34.
Solution
Since , then . Since , it follows that .
In the following exercises, determine the value of such that the function is continuous for the given value of .
35.
36.
Solution
37.
38.
Solution
In the following exercises, determine all horizontal and vertical asymptotes.
39.
40.
Solution
Horizontal: , Vertical:
41.
42.
Solution
Horizontal: , Vertical: none
In the following exercises, use the Intermediate Value Theorem to show that the given functions have an x-intercept in the given interval.
43. on the interval
44. on the interval
Solution
Since is continuous on [-2,-1] and and , then by IVT, there exists a root on the given interval.
45. A ball is thrown into the air and the vertical position is given by . Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.
46. A particle moving along a line has a displacement according to the function , where is measured in meters and is measured in seconds. Find the average velocity over the time period .
Solution
m/sec
In the following exercises, use the precise definition of limit to prove the limit.
47.
48.