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One-sided Limits
Muhammad Nadeem
School of Electrical Engineering &Computer Sciences
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One-sided Limit Let ƒ( x) is defined on an interval (a, b),
where a < b and ƒ( x) approaches
arbitrarily close to K as x approaches afrom within that interval, then ƒ has right-hand limit K at a. We write
K x f Lim =)( x
y
b
K
a x
a x→
M x f Limb x
=−
→
)(
Let ƒ( x) is defined on an interval (a, b),where a < b and ƒ( x) approachesarbitrarily close to M as x approaches b
from within that interval, then ƒ has Left-hand limit M at b. We write
x
y
ba x
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Limit Let ƒ ( x) is defined on an interval (a, b), where a < b and c is any
number within that interval. Function ƒ ( x) has a limit as x
approaches c if and only if it has left-hand and right-hand limitsat x=c and these one-sided limits are equal:
L x Lim =)( y
c x +
→
L x f Limc x
=−
→
)(
L x f Limc x
=→
)( xb
L
a x c x
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y
A function may fail to have a Limit at a point in its domain
1
=
≠=
0 0
0 1
)(
x
x x xg
y
≥
<=
1 1
1 0)(
x
x xU
x0 0
x
1)(0
=+
→
x f Lim x
0)(0
=−
→
x f Lim x
∞=+
→
)(0
x f Lim x
−∞=−
→
)(0
x f Lim x
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x + 52
Example: Find the limit
x x x −+→ 23
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5)3(2
)(
)52(52
2
2
3
3
23
+=
−
+
=
−
+
+
+
+
→
→
→ x x Lim
x Lim
x x
x Lim
x
x
x
6
11=
−
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x + 52
Example: Find the limit
x x x −−→ 23
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5)3(2
)(
)52(52
2
2
3
3
23
+=
−
+
=
−
+
−
−
−
→
→
→ x x Lim
x Lim
x x
x Lim
x
x
x
6
11=
−
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2)3( ++ x x
Example: Find the limit
22 ++−→ xm
x
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)2)(3(2)3( ++=
++
++
x x Lim
x x Lim
−+
=+
2 )2(
2 0
2 )2(
2
x x
x
x x
xSince
1
)3(2
+=
+=+
−→
−−
x Lim x
x
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2)3( ++ x x
Example: Find the limit
22 +−−→ xm
x
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)2)(3(2)3( ++−=
++
−−
x x Lim
x x Lim
−+
=+
2 )2(
2 0
2 )2(
2
x x
x
x x
xSince
1
)3(2
−=
+−=−
−→
−−
x Lim x
x x
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≤<
−
−
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2)()( )1(2
11 =+= −− →→ x x Lim x f Lim x x
222)( )2(11
=+=++
→→
x Lim x f Lim x x
2)( )3(1
=→
x f Lim x
Since LHL=RHL=2
)( )6(2
x f Lim x→
22
==−− →→ x x
4
1
2
1
4
2)( )5(
22
22
−=
+
−=
−
−=
+++→→→ x Lim
x
x Lim x f Lim
x x x
Doesn't exists
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≤<
≤
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x
y
1
0
2
3
1 2 3 41−2−3−
1−
2−
3−
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2)( )1(2
−=+
−→
x f Lim x
0)( )2(1
=−
−→ x f Lim
x
1)( )3(1
=+
−→
x f Lim x
→−→
)( )4(1
x f Lim x
0)( )5(0
=−
→
x f Lim x
3)( )6(0
=+
→
x f Lim x
Doesn't exists
→→
)( )7(0
x f Lim x
1)( )8(2
=−
→
x f Lim x
1)( )9(2
=+
→
x f Lim x
1)( )10(
2
=→
x f Lim
x
2)( )11(4
=−
→
x f Lim x
Doesn't exists
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Example: Let f is an odd function of x and
3)(0
=+
→
x f Lim x
Can you guess
If yes, write limit, if no, give reason.
?)(0
=−
→
x f Lim x
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Since f ( x) is odd function, that is
)()( x f x f −=−
and
Hence
3)(0
=+
→
x f Lim x
3)(0
−=−
→ x f Lim
x
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