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1 Functions and Limits

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Copyright © Cengage Learning. All rights reserved.

1.6 Calculating Limits Usingthe Limit Laws

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33

Calculating Limits Using the Limit Laws

In this section we use the following properties of limits,called the Limit Laws, to calculate limits.

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hese five laws can !e stated ver!ally as follows"

Sum Law 1. he limit of a sum is the sum of the

limits.

Difference Law 2. he limit of a difference is the

difference of the limits.

Constant Multiple Law 3. he limit of a constant times a

function is the constant times the

limit of the function.

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##


Product Law 4. he limit of a product is the productof the limits.

Quotient Law . he limit of a $uotient is the $uotient

of the limits %provided that the limit

of

the denominator is not &'.

(or instance, if f % x ' is close to L and g % x ' is close to M , it is

reasona!le to conclude that f % x ' ) g % x ' is close to L ) M .

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**

+ample -

Use the Limit Laws and the graphs of f and g in (igure - toevaluate the following limits, if they eist.

Fi!ure 1

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+ample -%a' / Solution

(rom the graphs of f and g we see that

and

herefore we have

0 - ) #%/-'

0 /4

%!y Law -'

%!y Law 3'

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11

+ample -%!' / Solution

2e see that lim x→- f % x ' 0 . ut lim x→- g % x ' does not eist

!ecause the left and right limits are different"

5o we can6t use Law 4 for the desired limit. ut we can use

Law 4 for the one7sided limits"

he left and right limits aren6t e$ual, so lim x→- 8f % x 'g % x '9

does not eist.

cont6d

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::

+ample -%c' / Solution

he graphs show that

and

ecause the limit of the denominator

is &, we can6t use Law #.

he given limit does not eist !ecause the denominator

approaches & while the numerator approaches a non;ero

num!er.

Fi!ure 1

cont6d

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-&-&


If we use the <roduct Law repeatedly with g % x ' 0 f % x ', weo!tain the following law.

In applying these si limit laws, we need to use two special

limits"

hese limits are o!vious from an intuitive point of view

%state them in words or draw graphs of y 0 c and y 0 x '.

<ower Law

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----


If we now put f % x ' 0 x in Law * and use Law 1, we getanother useful special limit.

A similar limit holds for roots as follows.

=ore generally, we have the following law.

>oot Law

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--


(unctions with the ?irect 5u!stitution <roperty are calledcontinuous at a.

In general, we have the following useful fact.

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-3-3


5ome limits are !est calculated !y first finding the left7 andright7hand limits. he following theorem says that a

two7sided limit eists if and only if !oth of the one7sided

limits eist and are e$ual.

2hen computing one7sided limits, we use the fact that the

Limit Laws also hold for one7sided limits.

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-4-4

he !reatest inte!er function is defined !y the largestinteger that is less than or e$ual to x . %(or instance,

' 5how that does not eist.

5olution"

he graph of the greatest integer function is shown in (igure *.

5ince for 3 x < 4,we have

+ample -&

Fi!ure 6

@reatest integer function

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-#-#

+ample -& / Solution

5ince for x < 3, we have

ecause these one7sided limits are not e$ual,

does not eist !y heorem -.

cont6d

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-*-*


The next two theorems give two additionalproperties of limits.

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he 5$uee;e heorem, which is sometimes called the5andwich heorem or the <inching heorem, is illustrated

!y (igure .

It says that if g % x ' is s$uee;ed !etween f

% x ' and h% x ' neara, and if f and h have the same limit L at a, then g is forced

to have the same limit L at a.

Fi!ure "

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