ページ1:

Assignment 1
Section 1.5
18. sketch the graph of an example of
a function f that satisfies all of
given
lim fox)=2
Condition
lim fox =0
lim fox) = 3
x-+0
x-94
f(4) =1✓
X-10°
lim fox) = f(0) = 2
*44
4
38. (a) Find the vertical asymptotes of
the function
x+1
y = 3x-2x²
19
No.
DATE
36. determine the the infinite limit.
lim x²-2x
x+2x²-4x+4
> lim
x(x-2)
*-2(x-2) (X-2)
=lim
*+2"
4-2
So we consider
x+1
lim
and
lim
x-40
3x-2x
x+1
3x-2
3x-2x²=0
+1
lim
3x-2x
2x²-3x=0
X-35
x(2x-3)=0
lim = ±
*0 3x-2x²
VA
x = 0
x = 3
x0,
Tim
**3*-2*
lim
x+1
**23*-2*2
Section 1.6
14. Evaluate the limit, (if exists)
Tim x²-4x
A-1
x²-3x-4
lim
-x--1"
x²-4x
x²-3x-4
lim
H
x(444)
*--5 (x-4)(x+1)
= 00
#
lim
x-3-1
4x
+238-4
lim
x(314)
*--+ (x-4)(x+1).
Hence, lim
X--1
=
x-4x
x-3x-4
=
does not exist.
-
16. lim
2x²+3x+1
**-*-2x-3
= lim (2x+x+)
(x-3)(3+)
*4-1
- lim
2X+1
x--1
x-J
2(-1)+1
-1-3

ページ2:

2810FZ
DATE
20. lim
+4-1
+- +-1
= lim (+11) (+²+1)
+ (+that+1)
22. lim Ju+1-3
Ze-n
= lim (+) (++1) (+=+1)
= lim
(4-11(++++1)
U-SI
(1+1)(1+7)
(14141)
262)
+
U-2
JAUH -3
U-2
= lim 40+1-9
1-32 (-2) (58 +3)
=lim
4(-1)
U-2 (U-23 (√40+T+3)
+3
√44+1 +3
40-8
32. lim
h-o
h
Im
x²-(x+4)
#4
h-po
lim
h+o
x²-x²-2xh-h²
x²(x+h)h
lim (-2x-h
-2x-0
-2x
x²(X+0)²
=
40. Prove that lim x [1 + sin ² ( 21 ) ] = 0
X48
-1 Sin (1
o≤ sin² (2) ≤ 1
1 ≤1+ sin² (27) ≤2
Sxx (1+ sin (2) ≤ 2√x
Tim Sx
==0
x-10
lim 2√x = 2500
since √ √ (1+sin² (21)
therefore, lim √x ≤ lim √x [1 + sin² ()] ≤ Tim 2√x
Hence,
x-x
0
slim
[1+sin² (2) ≤ 0
lim x[1+sin(-1) =0.
x-10

ページ3:

No.:
DATE.
42. Find the limit, if it exist. If the limit doesn't exist, explain why.
2x+12
lim
x716
Ix+bl
1x+b
{
*+6 + *+630 → *3-6
-(x+6); x <-66
case I ×4-6
lim
+
2x+12
=
lim
2x+12
x-6x+b
4778
x+6
= lim
77-6
20846)
446
=
2
Since
lim
2x+12
1x+6
lim
50. let
x
if *4
3
if x=1
g (x) =
2-x² if 12x ≤2
x-3 if x>2
b) Sketch the graph of
Y
4
-
2
+-2
-3
-4
g
Case I
x2-6
lim
2x+12
lim
19+×19-6x
2x+12
84-6-(x+6)
= lim 2(x+6)
(9+5)-76x
2x+12
**-61x+61
-2
Hence, limit does not exist.
a) evaluate each of the following, if it exists.
(islim gun
445
(ii) lim g(x)
371
(iii) g(1)
(iv) tim gcx
kat
(v) lim
= 1
lim
g(x) = 1, lim
A-1
gon = 1 lim gex =1
x+1
= 3
=
2-22-2
=
2-3
= 1
4
(vi) lim gcx)
+2
since lim g(x) lim
344
+9(x)
Hence, lim does not exist.
*-72

ページ4:

Section 1.8
NIVERSITY
No. :
DATE:
46. Find the value of a and b that make f continuous everywhere.
fox)
=
+24
x-2
if x<2
ax2 bx+3 if 25x <3
2x-a+b if x 3
fis continuous an
(-0,2) U(23) V (3,00)
So we only have to check at x = 2,3
at x= lim fox
=
Tim x²-4
lim (x-2)(x+2)
X-42
4-924-2
x-12
K-L
= 2+2=4
lim fox
tim
2x²-6x+3
9
42-26+3
continuous
lim fix lim f(x))
Xast
442-2b+3
at x=3
:
lim
f(x)
lim ax²-6x+3
= 92-36+3
475
lim fix
Continuous
lim fox
*45
92-36 +3
I'm 2x-2+b
*43+
6-2+b
=
lim
fox)
x-3*
ร
6-2+b
102-4b
=
9
2-20;
22 = 1
plug in anţ m ® ; 10(7)-46=9
5-46=9
b
1
b
-1
0.5

ページ5:

Section 2.4
44. find the limit
lim
sin 3x sin 3x
lim
x70
DATE 532
=
570
lim
040
Sin 3x sin 5x1
Sin 3x sin 3x xxx.1
x
X
-(lim sin 3x lim sin x)
×
(m3)(
lim
X-10
X x10
(5)
=
3.5
= 15
46. lim sin (x2)
X40
×
lim
X18
sin (x3
=
lim
X40
sin (x3.x
x²
lim
8714
Sin (x).
lim
1.0
X10
X

ページ6:

951540
SAT
No.
Section 3.4
DATE
2. a) Can the graph of y = f(x) intersect a vertical asymptote?
Can it intersect horizontal asymptote ? Illustrate by sketching graph.
• If the line x is a vertical asymptote
that's mean
lim f(x) = ∞ Or
X43
lim f(x) = &
e+x
Since x is approaching a, not exactly a
that is xa so x = 2 line will
not intersect the graph y= f(x)
YA
For horizontal asymptote, the graph
y=fix might intersect that he
for example
intersection!!
horizontal asymptote
.
0
v
x
a
How
many
the
graph
of y = f(x) have?
horizontal asymptotes can
Sketch graphs to illustrate the possibilities.
Since lim f(x) = L
8110
00-4x
and
lim f(x) = A
2
Hence, the maximum number of horizontal asymtote is 2
(In the case that ① and ② exist and L #A)
Y
A
horizontal
symptote
horizontal
symptote
Y
°
In the case that and exist and L = A,
the number of horizontal asymptote is 1
horizontal
asymptote
•Also the minimum
number of horizontal asymptote is of
(in the care that lim
fox) = ∞0-00
lim f(x)=0,0)
84x
>x
^
no horizontal
asymptote.

ページ7:

MASA
UNIVER
4. for the function g whose graph is given, state the following
alim g(x) = 2
008-8
d) limg(x)
=-00
b) lim g(x) = -1
86x
e) lim goo∞0
x+2*
The egation of the asymptotes
No.
DATE
c) linn gox =
080
Since lim g(x)
= 100
and lim 9 ) = 00
..
Hence, lim g(x) does not exist.
x-2
also
lim g(x)=0 ; SO x=0 is the vertical asymptote.
040
lim g(x) =2
776
0.5
and lim g(x) = -1 --> Hence, 2 and y = -1 are horizontal
17
asymptotes.
8. Evaluate the limit and justify each step by indicating the appropriate
Properties of limits.
lim
277
12x³-5x+2
=
lim
X (12-
5+ )
1+ 4x²+3x³
=
12-0 +0
0+0+3
=
3
= √4 = 2
AH
14. find the limit or show that it does not exist.
lim +=+
+ 2+++3+-5
lim F (-1)
++00
0-1
2+0-0
JF (2+)