M53 Lec1.1 Limits One-Sided1
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Limit of a Function and
One-sided limits
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 1 / 37
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For today
1 Limit of a Function: An intuitive approach
2 Evaluating Limits
3 One-sided Limits
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 2 / 37
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Introduction
Given a function f (x) and a R,what is the value of f at x near a,
but not equal to a?
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 4 / 37
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Illustration 1
Consider f (x) = 3x 1.
What can we say about values of f (x) for values of x near 1 but not equal to 1?
x f (x)0 1
0.5 0.50.9 1.7
0.99 1.970.99999 1.99997
x f (x)2 5
1.5 3.51.1 2.3
1.001 2.0031.00001 2.00003
Based on the table, as x gets closer and closer to 1, f (x) gets closer and closerto 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 5 / 37
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Illustration 1
x f (x)0 1
0.5 0.50.9 1.7
0.99 1.970.99999 1.99997
1 1 2 31
1
2
3
4
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 6 / 37
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Illustration 1
x f (x)2 5
1.5 3.51.1 2.3
1.001 2.0031.00001 2.00003
1 1 2 31
1
2
3
4
As x gets closer and closer to 1, f (x) gets closer and closer to 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 7 / 37
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Illustration 2
Consider: g(x) =3x2 4x + 1
x 1
=(3x 1)(x 1)
x 1= 3x 1, x 6= 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1, g(x) gets closer and closer to 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 8 / 37
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Illustration 3
Consider: h(x) =
3x 1, x 6= 10, x = 1
1 1 2 31
1
2
3
4
As x gets closer and closer to 1, h(x) gets closer and closer to 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 9 / 37
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Limit
Intuitive Notion of a Limita R, L R
f (x): function defined on some open interval containing a, except possibly at a
The limit of f (x) as x approaches a is L
if the values of f (x) get closer and closer to L as x assumes values getting closerand closer to a but not reaching a.
Notation:
limxa
f (x) = L
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 10 / 37
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Examples
f (x) = 3x 1
1 1 2 31
1
2
3
4limx1
(3x 1) = 2
Note: In this case, limx1
f (x) = f (1).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 11 / 37
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Examples
g(x) =3x2 4x + 1
x 1
1 1 2 31
1
2
3
4 limx1
3x2 4x + 1x 1 = 2
Note: Though g(1) is undefined,limx1
g(x) exists.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 12 / 37
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Examples
h(x) =
3x 1, x 6= 10, x = 1
1 1 2 31
1
2
3
4limx1
h(x) = 2
Note: h(1) 6= limx1
h(x).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 13 / 37
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Some Remarks
RemarkIn finding lim
xaf (x):
We only need to consider values of x very close to a but not exactly at a.
Thus, limxa
f (x) is NOT NECESSARILY the same as f (a).
We let x approach a from BOTH SIDES of a.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 14 / 37
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Some Remarks
If f (x) does not approach anyparticular real number as xapproaches a, then we saylimxa
f (x) does not exist (dne).
e.g.
H(x) =
1, x 0
0, x < 0
3 2 1 1 2 3
1
2
3
0
limx0
H(x) = 0? No.
limx0
H(x) = 1? No.
limx0
H(x) dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 15 / 37
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Limit Theorems
TheoremIf lim
xaf (x) exists, then it is unique.
If c R, then limxa
c = c.
limxa
x = a
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 17 / 37
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Limit Theorems
TheoremSuppose lim
xaf (x) = L1 and limxa g(x) = L2. Let c R, n N.
limxa
[ f (x) g(x)] = limxa
f (x) limxa
g(x) = L1 L2
limxa
[ f (x)g(x)] =(
limxa
f (x)) (
limxa
g(x))
= L1L2
limxa
[c f (x)] = c limxa
f (x) = cL1
limxa
f (x)g(x)
=limxa
f (x)
limxa
g(x)=
L1L2
, provided L2 6= 0
limxa
( f (x))n =(
limxa
f (x))n
= (L1)n
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 18 / 37
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Evaluate: limx1
(2x2 + 3x 4)
limx1
(2x2 + 3x 4) = limx1
2x2 + limx1
3x limx1
4
= 2(
limx1
x2)+ 3
(lim
x1x) lim
x14
= 2(
limx1
x)2
+ 3(
limx1
x) lim
x14
= 2(1)2 + 3(1) 4
= 5
In general:
RemarkIf f is a polynomial function, then lim
xaf (x) = f (a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 19 / 37
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Evaluate: limx2
4x3 + 3x2 x + 1x2 + 2
limx2
4x3 + 3x2 x + 1x2 + 2
=lim
x2(4x3 + 3x2 x + 1)
limx2
(x2 + 2)
=4(2)3 + 3(2)2 (2) + 1
(2)2 + 2
= 176
RemarkIf f is a rational function and f (a) is defined, then lim
xaf (x) = f (a).
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 20 / 37
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TheoremSuppose lim
xaf (x) exists and n N. Then,
limxa
n
f (x) = n
limxa
f (x),
provided limxa
f (x) > 0 when n is even.
limx3
3x 1 =
limx3
(3x 1) =
8 = 2
2
limx1
3
x + 4x 2 =
3
1 + 41 2 = 1
limx7/2
4
3 2x dne
limx2
x2 4 =? (for now)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 21 / 37
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Evaluate: limx3
(2x2
5x + 1
x3 x + 4
)3
limx3
(2x2
5x + 1
x3 x + 4
)3=
limx3 2x2
limx3
(5x + 1)
limx3
(x3 x + 4)
3
=
(18 4
28
)3=
18
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 22 / 37
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Consider: g(x) =3x2 4x + 1
x 1 . From earlier, limx1 g(x) = 2.
Can we arrive at this conclusion computationally?
Note that limx1
(3x2 4x + 1
)= 0 and lim
x1(x 1) = 0.
But when x 6= 1, 3x2 4x + 1
x 1 =(3x 1)(x 1)
x 1 = 3x 1.
Since we are just taking the limit as x 1,
limx1
3x2 4x + 1x 1 = limx1(3x 1) = 2.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 23 / 37
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DefinitionIf lim
xaf (x) = 0 and lim
xag(x) = 0 then
limxa
f (x)g(x)
is called an indeterminate form of type00
.
Remarks:
1 If f (a) = 0 and g(a) = 0, thenf (a)g(a)
is undefined!
2 The limit above MAY or MAY NOT exist.3 Some techniques used in evaluating such limits are:
Factoring
Rationalization
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 24 / 37
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Examples
Evaluate: limx2
x3 + 8x2 4
(00
)
limx2
x3 + 8x2 4 = limx2
(x + 2)(x2 2x + 4)(x + 2)(x 2)
= limx2
x2 2x + 4x 2
=4 + 4 + 42 2
= 3
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 25 / 37
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Examples
Evaluate: limx4
x2 162
x
(00
)
limx4
x2 162
x= lim
x4
x2 162
x 2 +
x
2 +
x
= limx4
(x2 16)(2 +
x)4 x
= limx4
(x 4)(x + 4)(2 +
x)4 x
= limx4
[(x + 4)(2 +
x)]
= (8)(4)
= 32
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 26 / 37
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Examples
Evaluate: limx1
3
x + 12x + 2
(00
)
limx1
3
x + 12x + 1
= limx1
3
x + 12x + 1
3x2 3
x + 1
3x2 3
x + 1
= limx1
x + 1
2(x + 1)( 3
x2 3
x + 1)
= limx1
1
2( 3
x2 3
x + 1)
=16
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 27 / 37
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Illustration 4
Consider: f (x) =
3 5x2, x < 1
4x 3, x 1
As x 1, the value of f (x) dependson whether x < 1 or x > 1. 4 3 2 1 1 2 3
3
2
1
1
2
3
4
0
As x approaches 1 through values less than 1, f (x) approaches 2.
As x approaches 1 through values greater than 1, f (x) approaches 1.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 29 / 37
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Illustration 5
Consider: g(x) =
x
2 1 1 2 3
1
1
2
0( )( )( )( )
Since there is no open interval I containing 0 such that g(x) is defined on I, wecannot let x approach 0 from both sides.
But we can say something about the values of g(x) as x gets closer and closer to0 from the right of 0.
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 30 / 37
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One-sided Limits
Intuitive DefinitionThe
limit of f (x) as x approaches a from the left is L
if the values of f (x) get closer and closer to L as the values of x get closer andcloser to a, but are less than a.
Notation:
limxa
f (x) = L
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 31 / 37
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One-sided Limits
Intuitive DefinitionThe
limit of f (x) as x approaches a from the right is L
if the values of f (x) get closer and closer to L as the values of x get closer andcloser to a, but are greater than a
Notation:
limxa+
f (x) = L
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 32 / 37
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Theoremlimxa
f (x) = L if and only if limxa
f (x) = limxa+
f (x) = L
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 33 / 37
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f (x) =
3 5x2, x < 1
4x 3, x 1
4 3 2 1 1 2 3
3
2
1
1
2
3
4
0
limx1
f (x) = limx1
(3 5x2) = 2
limx1+
f (x) = limx1+
(4x 3) = 1
limx1
f (x) dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 34 / 37
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Examples
g(x) =
x
2 1 1 2 3
1
1
2
0
Based on the graph,
limx0+
x = 0
limx0
x dne
limx0
x dne
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 35 / 37
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Examples
Let p(x) =
5 2x, x 2
x2 2x2x 4 , x > 2
limx2
p(x) = limx2
5 2x =
1 = 1
limx2+
p(x) = limx2+
x2 2x2x 4 = limx2+
x(x 2)2(x 2) = limx2+
x2= 1
limx2
p(x) = 1
limx3
p(x) = limx3
x2 2x2x 4 =
9 66 4 =
32
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 36 / 37
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Exercises
1 Evaluate: limx4
3x 123
2x + 1
2 Find limx1
f (x) given: f (x) =
x2 + 1x 1 , if x < 1
1
x + 5, if x 1
3 Evaluate: limx2/3
(6x 4
3x2 + 4x 4 +1
3x + 2
)
Institute of Mathematics (UP Diliman) Limit of a Function and One-sided limits Mathematics 53 37 / 37
Limit of a Function: An intuitive approachEvaluating LimitsOne-sided Limits