Module 2 Review Problems

True or False. In the following exercises (1-4), justify your answer with a proof or a counterexample.

1. A function has to be continuous at x=a if the limx→af(x) exists.

2. You can use the quotient rule to evaluate limx→0sinxx.

3. If there is a vertical asymptote at x=a for the function f(x), then f is undefined at the point x=a.

4. If limx→af(x) does not exist, then f is undefined at the point x=a.

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

  1. limx→−1f(x)
  2. limx→1f(x)
  3. limx→0+f(x)
  4. limx→2f(x)

A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).

In the following exercises (6-15), evaluate the limit algebraically or explain why the limit does not exist.

6. limx→22x2−3x−2x−2

7. limx→03x2−2x+4

8. limx→3x3−2x2−13x−2

9. limx→π2cotxcosx

10. limx→−5x2+25x+5

11. limx→23x2−2x−8x2−4

12. limx→1x2−1x3−1

13. limx→1x2−1√x−1

14. limx→44−x√x−2

15. limx→41√x−2

In the following exercises (16-17), use the squeeze theorem to prove the limit.

16. limx→0x2cos(2πx)=0

17. limx→0x3sin(πx)=0

18. Determine the domain such that the function f(x)=√x−2+xex is continuous over its domain.

In the following exercises (19-20), determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.

19. f(x)={x2+1 if x>c2x if x≤c

20. f(x)={√x+1 if x>−1x2+c if x≤−1

In the following exercises (21-22), use the precise definition of limit to prove the limit.

21. limx→1(8x+16)=24

22. limx→0x3=0

23. A ball is thrown into the air and the vertical position is given by x(t)=−4.9t2+25t+5. 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 x(t)=t2−2t+4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t=[0,2].

25. From the previous exercises, estimate the instantaneous velocity at t=2 by checking the average velocity within t=0.01 sec.