True or False. In the following exercises, 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.
- limx→−1f(x)
- limx→1f(x)
- limx→0+f(x)
- limx→2f(x)
In the following exercises, 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→π/2cotxcosx
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, 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, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous.
19. f(x)={x2+1ifx>c2xifx≤c
20. f(x)={√x+1ifx>−1x2+cifx≤−1
In the following exercises, 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.
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