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Calculus 1

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1
By Gökhan Bilhan
Calculus 1
(Week 4)-Limits Part1
Denition(Limit) We write limx→c f (x) = L, if the value f (x) is close to the single real number L
whenever x is close but NOT EQUAL to c.
Remarks
(1) x must approach to c from left and right.
(2) c, need not to be in the domain.
(3) x ̸= c, always.
Example
limx→0 |x|
limx→0
|x|
x
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By Gökhan Bilhan
Denition (One Sided Limits)
We write limx→c− f (x) = K and say, limit from left (or left hand limit) if x approaches to c
from left. In that case x < c.
We write limx→c+ f (x) = K and say, limit from right (or right hand limit) if x approaches to c
from right. In that case x > c.
Theorem(Existence of a Limit) limx→c f (x) = A if and only if limx→c
+
f (x) = limx→c− f (x) = A
At open holes, function is not dened.
At closed (dotted) holes function is dened.
limx→−1− f (x) =?
limx→−1+ f (x) =?
limx→−1 f (x) =?
limx→1− f (x) =?
limx→1+ f (x) =?
limx→1 f (x) =?
limx→2− f (x) =?
limx→2+ f (x) =?
limx→2 f (x) =?
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By Gökhan Bilhan
Theorem (Properties of Limits)
Let f and g be two functions and assume that limx→c f (x) = L and limx→c f (x) = M where L
and M are real numbers (i.e. both limits exist). Then
1-) limx→c k = k for any constant k .
2-) limx→c x = c.
3-) limx→c (f (x) + g(x)) = limx→c f (x) + limx→c g(x) = L + M .
4-) limx→c (f (x) − g(x)) = limx→c f (x) − limx→c g(x) = L − M .
5-) limx→c (f (x).g(x)) = limx→c f (x).limx→c g(x) = L.M .
6-) limx→c kf (x) = klimx→c f (x) = kL.
f (x)
limx→c f (x)
L
=
=
.
g(x)
lim
M √
x→c g(x)
√
√
8-) limx→c n f (x) = n limx→c f (x) = n L note that L > 0 if n is even.
7-) limx→c
Examples
1-) limx→3 (x2 − 4x) =?
√
2-) limx→−1 2x2 + 3 =?
√
limx→−1 2x3 + 1 =?
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By Gökhan Bilhan
{
x2 + 1, if
3-) f (x) =
x − 1, if
(a) limx→2− f (x) =?
(c) limx→2 f (x) =?
x<2
x>2
(b) limx→2+ f (x) =?
(d) f (2) =?
Exercises
1. Answer the following by looking at the following graph
a-) limx7→4− f (x)
b-) limx7→4+ f (x)
c-) limx7→4 f (x)
d-) f (4) =?
e-) Is it possible to dene f (4) so that limx7→4 f (x) = f (4)
f-) limx7→2− f (x)
g-) limx7→2+ f (x)
h-) limx7→2 f (x)
k-) f (2) =?
l-) Is it possible to dene f (2) so that limx7→4 f (x) = f (2)
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By Gökhan Bilhan
2. Same Question
3. Let f (x) =
3x2 + 2x − 1
, nd
x2 + 3x + 2
a) limx→−3 f (x) =?
c) limx→2 f (x) =?
b) limx→−1 f (x) =?
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By Gökhan Bilhan
x2 + x − 6
4. Let f (x) =
, nd
x+3
a) limx→−3 f (x) =?
b) limx→0 f (x) =?
c) limx→2 f (x) =?
{
x2 ,
if
5. Let f (x) =
x − 1, if
a) limx→1+ f (x) =?
c) limx→1 f (x) =?
x<1
x>1
b) limx→1−1 f (x) =?
d) f (1).
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By Gökhan Bilhan
{
1 + mx, if
6. Let f (x) =
4 − mx, if
x≤1
where m is a constant.
x>1
a) Graph f for m = 1.
b) limx→1− f (x) =?
c) limx→1+ f (x) =?
d) Find m so that limx→1− f (x) = limx→1+ f (x)
7. limh→0
[3(a + h) − 2] − (3a − 2)
=?
h
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By Gökhan Bilhan
Denition (Indeterminate Form) If limx→c f (x) = 0 and limx→c g(x) = 0,then limx→c
to be indeterminate or more specically a
Example limx→2
x2 − 1
=?
x−1
0
indeterminate form.
0
f (x)
is said
g(x)
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By Gökhan Bilhan
Example limx→−1
x|x + 1|
=?
x+1
Example limx→−1,00000000001
x|x + 1|
=?
x+1
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By Gökhan Bilhan
Example limx→1,000001
Example If f (x) =
Example limx→−1
√
|x|
=?
x
x, then nd limh→0
f (2 + h) − f (2)
=?
h
x−1
=?(Can we solve it yet?)
x+1
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By Gökhan Bilhan
Exercises
√
√
1. limh→0
a+h−
h
2. limx→a
x2 − a2
=?
x−a
3. limx→−2
a
=?
2x2 + 3x − 2
=?
x+2
1 1
−
4. limx→2 x 2 =?
x−2
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By Gökhan Bilhan
√
5. limx→−2
6. limx→2
x2
=?
x
x2
=?
x
x
x
7. limx→0 =?
√
limx→0
limx→0
limx→2
x2
=?
x
x2
=?
x
x
=?
x
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By Gökhan Bilhan
(Week 4)-Limits(Exercises)
Exercises
1. limx→3
A) 23
2. limx→2
A)-1
x3 − 3x2
=?
x2 − 3
B) 12
C)0
D)3
E)6
x3 − 8x + 8
=?
x4 − 4x
B)
−1
7
C)0
D)1
E)2
√
3− a−8
exists.
3. What is "a" to say the limit limx→2
x−2
A)12
B)11
4. limx→1− (
A)1
5. limy→x
A)0
C)5
D)3
E)2
|1 − x|
+ x) =?
1−x
B)2
C)−1
D)−2
E)0
y 3 − x3
=?
y 2 − x2
3
B) x
2
C)2x
2
D) x
3
E)2
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By Gökhan Bilhan
1
8 =?
6. limx→ 1
1
2
2
x −
4
x3 −
A)− 43
B)− 34
C) 43
√
3
x−4
7. limx→64 √
=?
x−8
A)0
B) 13
D) 18
E) 21
1
Hint:( x 6 = t )
C) 23
D) 32
E)3

 |x|
, if
x ̸= 0
8. Let f (x) =
and
x
3,
if
x=0
if limx→0+ f (x) = a and limx→0− f (x) = b , then what is a − b?
A)−2
B)−1
C)0
D)1
E)2
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By Gökhan Bilhan

2

if
x ,
9. Let f (x) = 3,
if


x + a, if
x<3
x=3
x>3
For the above function f , what is a to say the limit at x = 3 exists?
A)4
B)6
C)7
10. Look at the graphs
D)8
E)9
By Gökhan Bilhan
Answers:
1-) C , 2-) C , 3-)B , 4-)B , 5-)B , 6-)C , 7-)B , 8-)E , 9-)B
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