2.R: Chapter 2 Review Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
True or False. In 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.
- Answer
- False, since we cannot have limx→0x=0 in the denominator.
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.
- Answer
- False. A jump discontinuity is possible.
5) Using the graph, find each limit or explain why the limit does not exist.
a. limx→−1f(x)
b. limx→1f(x)
c. limx→0+f(x)
d. limx→2f(x)
In exercises 6 - 15, evaluate the limit algebraically or explain why the limit does not exist.
6) limx→22x2−3x−2x−2
- Answer
- 5
7) limx→03x2−2x+4
8) limx→3x3−2x2−13x−2
- Answer
- 8/7
9) limx→π/2cotxcosx
10) limx→−5x2+25x+5
- Answer
- DNE
11) limx→23x2−2x−8x2−4
12) limx→1x2−1x3−1
- Answer
- 2/3
13) limx→1x2−1√x−1
14) limx→44−x√x−2
- Answer
- −4
15) limx→41√x−2
In exercises 16 - 17, use the squeeze theorem to prove the limit.
16) limx→0x2cos(2πx)=0
- Answer
- Since −1≤cos(2πx)≤1, then −x2≤x2cos(2πx)≤x2. Since limx→0x2=0=limx→0−x2, it follows that 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.
- Answer
- [2,∞]
In 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 exercises 21 - 22, use the precise definition of limit to prove the limit.
21) limx→1(8x+16)=24
22) limx→0x3=0
- Answer
- δ=3√ε
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].
- Answer
- 0 m/sec
25) From the previous exercises, estimate the instantaneous velocity at t=2 by checking the average velocity within t=0.01 sec.