M53 Lec1.1 Limits One-Sided1

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

For today

1 Limit of a Function: An intuitive approach

2 Evaluating Limits

3 One-sided Limits


Introduction

Given a function f (x) and a R,what is the value of f at x near a,

but not equal to a?


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.


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


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.


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.


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.


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


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).


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.


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).


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.


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


Limit Theorems

TheoremIf lim

xaf (x) exists, then it is unique.

If c R, then limxa

c = c.

limxa

x = a


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


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).


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).


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)


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


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.


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


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


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


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


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.


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.


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


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


Theoremlimxa

f (x) = L if and only if limxa

f (x) = limxa+

f (x) = L


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


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


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


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

)


Limit of a Function: An intuitive approachEvaluating LimitsOne-sided Limits

Documents

M53 Lec1.1 Limits One-Sided1