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Mathematics LibreTexts

2.R: Chapter 2 Review Exercises

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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 limxaf(x) exists.

2) You can use the quotient rule to evaluate limx0sinxx.

Answer
False, since we cannot have limx0x=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 limxaf(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. limx1f(x)

b. limx1f(x)

c. limx0+f(x)

d. limx2f(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 exercises 6 - 15, evaluate the limit algebraically or explain why the limit does not exist.

6) limx22x23x2x2

Answer
5

7) limx03x22x+4

8) limx3x32x213x2

Answer
8/7

9) limxπ/2cotxcosx

10) limx5x2+25x+5

Answer
DNE

11) limx23x22x8x24

12) limx1x21x31

Answer
2/3

13) limx1x21x1

14) limx44xx2

Answer
4

15) limx41x2

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

16) limx0x2cos(2πx)=0

Answer
Since 1cos(2πx)1, then x2x2cos(2πx)x2. Since limx0x2=0=limx0x2, it follows that limx0x2cos(2πx)=0.

17) limx0x3sin(πx)=0

18) Determine the domain such that the function f(x)=x2+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 xc

20) f(x)={x+1,if x>1x2+c,if x1

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

21) limx1(8x+16)=24

22) limx0x3=0

Answer
δ=3ε

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)=t22t+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.


2.R: Chapter 2 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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