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Lesson 4_One-Sided Limits - Copy

May 03, 2017

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Page 1: Lesson 4_One-Sided Limits - Copy

LIMITSOF

FUNCTIONS

Page 2: Lesson 4_One-Sided Limits - Copy

DEFINITION: LIMITS The most basic use of limits is to describe how a function behaves as the independent variable approaches a given value. For example let us examine the behavior of the function for x-values closer and closer to 2. It is evident from the graph and the table in the next slide that the values of f(x) get closer and closer to 3 as the values of x are selected closer and closer to 2 on either the left or right side of 2. We describe this by saying that the “limit of is 3 as x approaches 2 from either side,” we write

1xx)x(f 2

1xx)x(f 2

31xxlim 2

2x

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2

3

f(x)

f(x)

x

y

1xxy 2

x 1.9 1.95 1.99 1.995 1.999 2 2.001 2.005 2.01 2.05 2.1

F(x) 2.71 2.852 2.97 2.985 2.997 3.003 3.015 3.031 3.152 3.31

left side right side

O

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This leads us to the following general idea.

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EXAMPLEUse numerical evidence to make a conjecture about the value of .

1x1xlim

1x

Although the function is undefined at x=1, this has no bearing on the limit. The table shows sample x-values approaching 1 from the left side and from the right side. In both cases the corresponding values of f(x) appear to get closer and closer to 2, and hence we conjecture that and is consistent with the graph of f.

1x1x)x(f

21x

1xlim1x

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x .99 .999 .9999 .99999 1 1.00001 1.0001 1.001 1.01

F(x) 1.9949 1.9995 1.99995 1.999995 2.000005 2.00005 2.0005 2.004915

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THEOREMS ON LIMITS

Our strategy for finding limits algebraically has two parts:•First we will obtain the limits of some simpler function•Then we will develop a list of theorems that will enable us to use the limits of simple functions as building blocks for finding limits of more complicated functions.

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We start with the following basic theorems, which are illustrated in Fig 1.2.1

axlim b kklim a numbers. real be k and a Let Theorem 1.2.1

axax

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Page 10: Lesson 4_One-Sided Limits - Copy

33lim 33lim 33lim example, For

a. of values all for ax as kf(x) why explains whichvaries, x as k at fixed remain

f(x) of values the then function, constant a is k xf If

x0x-25x

Example 1.

xlim 2xlim 0xlim

example, For . axf that true be also must it ax then x, xf If

x-2x0x

Example 2.

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The following theorem will be our basic tool for finding limits algebraically.

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This theorem can be stated informally as follows:

a) The limit of a sum is the sum of the limits.b) The limit of a difference is the difference of the limits.c) The limits of a product is the product of the limits.d)The limits of a quotient is the quotient of the limits, provided the limit of the denominator is not zero.e) The limit of the nth root is the nth root of the limit.

•A constant factor can be moved through a limit symbol.

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5x2lim .14x

12x6lim .23x

)2x5(x4lim .33x

EXAMPLE : Evaluate the following limits.

3158

5)4(2

5limxlim2

5limx2lim

4x4x

4x4x

612-18

12)3(6

12limx6lim3x3x

13

131 2)3(534

2limxlim5xlim4lim

2limx5limxlim4lim

2x5limx4lim

3x3x3x3x

3x3x3x3x

3x3x

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4x5x2lim .4

5x

3

3x6x3lim .5

3x1x8lim .6

1x

2110

42552

4limxlim5

x lim2

4limx5lim

x2 lim

5x5x

5x

5x5x

5x

337515633

6limxlim3

6limx3lim

6x3lim

33

3

3x3x

3

3x3x

3

3x

23

49

3x1x8lim

1x

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OR

When evaluating the limit of a function at a given value, simply replace the variable by the indicated limit then solve for the value of the function:

22

3lim 3 4 1 3 3 4 3 1

27 12 138

xx x

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EXAMPLE: Evaluate the following limits.

2x8xlim .1

3

2x

Solution:

00

088

2282

2x8xlim

33

2x

Equivalent function:

(indeterminate)

2x

4x2x2xlim2

2x

124444222

4x2xlim2

2

2x

122x8xlim

3

2x

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Note: In evaluating a limit of a quotient which reduces to , simplify the fraction. Just remove the common factor in the numerator and denominator which makes the quotient . To do this use factoring or rationalizing the numerator or denominator, wherever the radical is.

00

00

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x22xlim .2

0x

Solution:

Rationalizing the numerator:

(indeterminate)00

0220

x22xlim

0x

22xx22xlim

22x22x

x22xlim

0x0x

42

221

221

22x1lim

22xxxlim

0x0x

42

x22xlim

0x

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9x427x8lim .3 2

3

23x

Solution:

By Factoring:

(indeterminate)32

3

3

22

38 278 27 27 27 02lim4 9 9 9 034 9

2

x

xx

3

232

9236

234

3x29x6x4lim

3x23x29x6x43x2lim

2

2

23x

2

23x

223

23

29

627

33999

223

9x427x8lim 2

3

23x

Page 20: Lesson 4_One-Sided Limits - Copy

5x3x2xlim .4 2

3

2x

Solution:

33

222

2 2 2 32 3lim5 2 5

8 4 34 5

159

153

x

x xx

315

5x3x2xlim 2

3

2x

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DEFINITION: One-Sided Limits

The limit of a function is called two-sided limit if it requires the values of f(x) to get closer and closer to a number as the values of x are taken from either side of x=a. However some functions exhibit different behaviors on the two sides of an x-value a in which case it is necessary to distinguish whether the values of x near a are on the left side or on the right side of a for purposes of investigating limiting behavior.

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Consider the function

0x ,10x ,1

xx

)x(f

1

-1

As x approaches 0 from the right, the values of f(x) approach a limit of 1, and similarly , as x approaches 0 from the left, the values of f(x) approach a limit of -1.

1xx

lim and 1xx

lim

,symbols In

oxox

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This leads to the general idea of a one-sided limit

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EXAMPLE:

xx

)x(f 1. Find if the two sided limits exist given

1

-1

exist. not does xx

lim or

exist not does itlim sided two the thenxx

limxx

lim the cesin

1xx

lim and 1xx

lim

ox

oxox

oxox

SOLUTION

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EXAMPLE:2. For the functions in Fig 1.1.13, find the one-sided limit and the two-sided limits at x=a if they exists.

The functions in all three figures have the same one-sided limits as , since the functions are Identical, except at x=a.

ax

1)x(flim and 3)x(flimare itslim These

axax

In all three cases the two-sided limit does not exist as because the one sided limits are not equal. ax

SOLUTION

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3. Find if the two-sided limit exists and sketch the graph of

2

6+x if x < -2( ) =

x if x -2g x

4 26

x6lim)x(glim.a2x2x

4

2-

xlim)x(glim.b

2

2

2x2x

4)x(glim or4 to equal is and exist itlim sided two the then

)x(glim)x(glim the cesin

2x

2x2x

SOLUTION

EXAMPLE:

Page 29: Lesson 4_One-Sided Limits - Copy

x-2-6 4

y

4

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4. Find if the two-sided limit exists and sketch the graph of

2

2

3 + x if x < -2( ) = 0 if x = -2

11 - x if x > -2

f x

SOLUTION

7 23

x3lim)x(flim.a

2

2

2x2x

7 2-11

x11lim)x(flim.b

2

2

2x2x

7)x(flim or7 to equal is and exist itlim sided two the then

)x(flim)x(flimthe cesin

2x

2x2x

EXAMPLE:

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graph. the sketch and

,exist f(x) lim if eminerdet ,4x23)x(f If .52x

3

4223

4x23lim )x(flim .a2x2x

3

4223

4x23lim )x(flim .b2x2x

3)x(flim or3 to equal is and exist itlimsided two the then

)x(flim)x(flimthe cesin

2x

2x2x

SOLUTION

EXAMPLE:

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f(x)

x

(2,3)

2

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DEFINITION: LIMITS AT INFINITY

The behavior of a function as x increases or decreases without bound is sometimes called the end behavior of the function.

)x(f

If the values of the variable x increase without bound, then we write , and if the values of x decrease without bound, then we write .

xx

For example ,

0x1lim and 0

x1lim

xx

Page 34: Lesson 4_One-Sided Limits - Copy

x

x

0x1lim

x

0

x1lim

x

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In general, we will use the following notation.

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Fig.1.3.2 illustrates the end behavior of the function f when L)x(flim or L)x(flim

xx

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EXAMPLEFig.1.3.2 illustrates the graph of . As suggested by this graph,

x

x11y

ex11lim

and ex11lim

x

x

x

x

Page 38: Lesson 4_One-Sided Limits - Copy

EXAMPLE

6x32xlim .4

x311x2x5lim .3

5x2xx4lim .2

8x65x3lim .1

2

x

23

x

3

2

x

x

336

x

36

x

xx5xlim .6

x5xlim .5