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. 2. 3. 4.  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. #### Solution

5

7. 8. #### Solution 9. 10. #### Solution

DNE

11. 12. #### Solution 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. 