2. Limits

# Chapter 2 Review Exercises

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, find each limit or explain why the limit does not exist.

1, starting at the open circle at (1,1).”>

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

6.

5

7.

8.

9.

10.

DNE

11.

12.

13.

14.

#### Solution

−4

15.

In the following exercises, use the squeeze theorem to prove the limit.

16.

#### Solution

Since , then . Since , it follows that .

17.

18. Determine the domain such that the function is continuous over its domain.

#### Solution

In the following exercises, determine the value of such that the function remains continuous. Draw your resulting function to ensure it is continuous.

19.

20.

#### Solution

In the following exercises, use the precise definition of limit to prove the limit.

21.

22.

#### Solution

23. 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.

24. 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

25. From the previous exercises, estimate the instantaneous velocity at by checking the average velocity within sec.

## License

Chapter 2 Review Exercises by OSCRiceUniversity is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.