MATH 161
groupwork III
25th May 2017
1. For the graph y = f(x) in the figure below, arrange the following numbers from smallest to largest:
A. The slope of the graph at A.
B. The slope of the graph at B.
C. The slope of the graph at C.
AB. The slope of the line AB.
0. The number 0.
1. The number 1.
Explain the positions of the number 0 and the number 1 in your ordering. Any unclear answers will be counted as
incorrect.
y
C
B
A
y= x
<
Answer:
y = f (x)
<
<
<
x\
<___________
2. Estimate the following limit by substituting appropriate values of x. (Show your results in tabular form. Give your
answer to two decimal places.)
5x  5
lim
x1
x 1
3. Use the graph below to find approximate values for each of the following limits (if they exist).
(a) lim f ( x)
x  4
(b) lim f ( x)
x  1
(c) lim f ( x)
x 2
(d ) lim f ( x)
x 6
2
4. State the Squeeze Theorem [aka, Sandwich Theorem, Pinching Theorem, Two Gendarmes Theorem, Two Policemen
and a Drunk Theorem]. Using actual functions, give an example of how it might be employed.
5. Calculate each of the following limits or explain why the limit does not exist. Justify each answer. If you use the
Squeeze Theorem, be precise.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
lim
x 
sin x
x
x 4  11
lim
x 3  x  34
lim x8 cos4 (1 / x )
x 0
sin ln( 5  x ) 
x 
x 1
x 4  16
lim
x 2
x2
lim
 2017 
lim x 4 cos 

x 0
 x 
lim x 2 e sin(1/ x )
x 0
6. Compute each of the following limits or explain why the limit fails to exist. Justify your reasoning. (A calculator
solution earns only partial credit.)
(a)
(b)
x 2 ( x  2)( x  3)
lim
x 2
| x 2|
x 
 sin 5 x
lim 


x 0
sin 4 x 
 x
x
cos 9 x
(c)
lim
(d)
lim sin
x 0
x 0
1
x
(Hint: Make the substitution z = 5x.)
3
(e)
 sin x
 13  
lim 
 cos   
x 
 x 
 x
7. Without using a calculator, compute
tan 3 2017 x 
lim
x 0
tan 3 4 x 
Note: You will need the following significant result which we shall soon prove:
sin 𝑥
𝑥→0 𝑥
lim
8. Consider the rational function F defined by
F ( x) 
15 x 3  x 2  6 x
6x2  x  2
(a) Where is F undefined? (Hint: Your answer should consist of two x values.)
(b) Let p denote the smaller of the two numbers found in part (a). Is it possible to extend F to a function that is
continuous at x = p? Explain.
(c) Let q denote the larger of the two numbers found in part (a). Is it possible to extend F to a function that is
continuous at x = q? Explain.
9. Compute (showing your work):

5
sin   

1

tan 5 x
 x
lim 
e x 
x 0 
x
ln x 




10. Does the following limit exist? Explain why or why not.
lim
x 13
| 3 x  39 |
x  13
11. Calculate each of the following limits. Briefly justify each answer.
x8  1
(a) lim 3
x1 x  1
Note: You will need to factor x3 – 1
(b) lim x 4 cos(1/ x)
x0
(c)
lim x sin( 1 / x)
x 
12. Which, if any, of the following functions possesses a limit as x→0? Briefly explain.
(a) y = 1/x
4
(b) y = sin (1/x)
(c)
y = ln x
(d) y = exp(1/x)
(e) y 
13.
cos x
x
(a) lim
tan 3x
x
(b) lim
sin 8 x
sin 11x
x 0
x 0
(c )
lim
x 0


sin  x  
2

(3x  1) 75
14. Does there exist a continuous extension to the curve
g ( x) 
3x 2  4 x  1
x4 1
at x = 1? If so, find it; if not explain!
15. Classify the type of discontinuity [removable, jump, infinite, essential] for each of the following:
(a)
(b)
(c )
(c )
|x|
at x  0
x
1
at x  0
x
1
at x  3
x  34
cos
2 x 2  9 x  35
at x  7
x7
12 x 2  7 x  10
(d )
at x  5
x5
16. Carefully state the Intermediate Value Theorem. Let f(x) = 7 + 2x – x3 be defined on the interval [1, 3].
(a) Explain why f must assume the value 0 somewhere on this interval.
(b)
Must f assume the value -13 on the interval [1, 3]? Does the Theorem imply that f must assume the value
9.3 on the interval [1, 3]?
5
 tan 5x
x
3
lim 
 x csc  x sin  .
3
x 0
2
x
 tan 2 x
3
17. Compute
18. Compute
lim
x 0
sin ax
.
sin bx
Show your work.
Have you made any assumptions about the constants a and b?
19. [University of Michigan] A portion of the graph of a function f is shown below.
y
y = f (x )
3
2
1
0
−1
1
2
3
4
5
6
7
8
9
10
x
−2
−3
(a) Find all values c in the interval 0 < c < 10 for which lim 𝑓(𝑥) does not exist. If none, write NONE.
𝑥→𝑐
(b) Give all values c in the interval 0 < c < 10 for which f(x) is not continuous at c. If there are none, write NONE.
(c) With f as shown in the graph above, define a function g by the formula
where A and B are nonzero constants.
Find values of A and B so that both of the following conditions hold.

g(x) is continuous at x = 0.
.
If no such values exist, write NONE in the answer blanks. Be sure to show your work or explain your
reasoning.
"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen.
"When I was younger, I always did it forhalf an hour a day.
Why, sometimes I've believed as many as six impossible things before
breakfast."
- Lewis Carroll, Alice in Wonderland