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. **

#### 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.