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Unformatted text preview: S ‘o ’{ﬁmb Sample Exam: Test One (2.1 — 2.5, 4.5) You must show all of your work to receive credit. No partial credit will be awarded
to Multiple Choice answers. Multiple Choice: (25 points) 1. For which of the following functions f (if any) does limf (x) fail to exist? x—rl E. limf(x) exists for all of the functions above. xal 2. Find the iimit lim 5‘" x X—ém x if it exists and give a reason to support your answer.
A. Does not exist; Indeterminate form 0%: B. L21; Special Limit
b C. Does not exist' Osci ting behavior
I. L =0; Sueeze Theorem E. Does not exist; Unbounded behavior 3. Find limLEZm—q, if it exists. 6—H) A.0 Be? D. m2
E. Does not exist X 4. Find any x—values at which I” (x) = is not continuous and identify x2 — 2x
each discontinuity‘as removable or nonremovable. A. only it = 2 (nonremovable) B. x = 0 (nonremovable ' x = u emovable)
C. x = 0 (removable); x = 2 (nonremovable) D. x = 0 (nonremova E. only x = 0 (removable) ’
5. Find the one—sided limit: lim “7
x—>4+ 4— x
°°
B. oo
C. —1
D. 0
7
E. 4 Free Response Find the following limits (25 points): 1. limsma "" SE :53
B—m 6 .i—rv \ . x2—25 _ y g +5)
3 — QM V“ = ,am. “5:51
Koﬁ 3,(5 $915 4. lim 3x5 —6x7 +x3—9x+1. _. 94> 'r—ﬁ—cn ..   .l 2 .
3x2—1, x;<—l Law 8! 4‘}: E] 5. lirqu(x)forf(x)= x, —15x<1 3?!
3x2+L x21 6. Forthe function pictured below (9 points): a. Find all the discontinuities of the function. b. Determine which condition of continuity is violated at each
discontinuity. 0. Determine whether the discontinuity is removable or nonremovable. : 0
La“ Rag) bgri .H ', 2914 Cf“) 7/4101.) “ 'PC3Jt3ME
4° K41 ' ‘ :  ..
1: WV abie .
Nan vcJI'C Qua—neat": It
2
—1 +1
7. Let f “kw. Find all asymptotes of this function. Provide limit
2x(x—l)(x+2)
statements to justify your answers (6 points).
V.A. Limit Statement:
X110 Damn 903:59 6“. QAan‘P’L")'°o
x—vc"' “4°
—....7. ﬂow. £Ct)=°° at Le... godzoo
viaNi" alt9‘3
H.A. Limit Statement:
1
_ l a _
‘:__ “gamma—3M QWQW 2
a_ 2“ “ ‘ ' 349° 3‘4““:
8. Create a function, g that possesses the following characteristics (3 points): a. g (t) has a vertical asymptote at x = 2
b. limg(t) =1 C. g (0) does not exist, but limg (t) =1 r—>0 9. Find a 5> Oto prove that lim3x—6 = —3. (2 points) xal 13}L~—C~3)]<E’ lx'ljz‘g 10. Determine whether the following statements are True or False. If false,
change the statement so that it is true (5 points) F1152. Bonuslllll! Oscillating Behavior is one of the reasons why a limit does not
exist. limf(x) = oo means that he limit exist at x = c and is equal tOoo. if f(x) is continuous on [a, b], andf(a) Sk s f(b), then there is a c
in [a, b] such thatf(c)=k. 3x3 x i I
= h a slant mféte 794! l  ~
f y/ 7% 1 92’ 25y y x 3w
If ling f (x) = oo, the ' y 2c is a horizontal asymptote of the function. 1. Explain why the function must be continuous for the Intermediate Value
Theorem to be valid (drawing a picture might be easier). I—cosx 2. Prove that lim =0: I—)0 x (Hint: Try an algebraic manipulation and remember the Pythagorean
identity!) ...
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 Summer '11
 MarioBorha
 Calculus

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