limites

1http://www.damasorojas.com.veDr.DMASOROJAS

UNIVERSIDAD POLITCTICA TERRITORIAL

JOS ANTONIO ANZOTEGUI MATEMTICAS PARA INGENIEROS

DEFINICINDELADERIVADA.La derivada de la funcin en un punto equivale a la pendiente de la recta tangente.Definida de manera informal, la derivada de una funcin en un punto es el valor delapendientedelarectatangenteendichopunto.Lapendienteestdadaporlatangentedelnguloqueforma larectatangentea lacurva(funcin)conelejede lasabscisas,enesepunto.Laderivadadeunafuncinmideelcoeficientedevariacindedichafuncin.Esdecir,proveeuna formulacinmatemticade lanocindelcoeficientedecambio.Elcoeficientede cambio indica lo rpidoque crece (odecrece)una funcin enunpunto(razndecambiopromedio)respectodelejedeunplanocartesianodedosdimensiones

Definicin. Sea una funcin definida en un intervalo abierto que contiene a a. La

derivadade ena,denotadapor (a),estdadapor0

( ) ( )( )h

f a h f af a Lmh

+ = , siestelmiteexiste.

Sieste lmiteexiste,decimosque esdiferenciableena.Encontrar laderivadase llamaderivacin;lapartedeclculoasociadaconladerivadasellamaclculodiferencial.

Ladiferenciabilidadimplicacontinuidad.Siunacurvatienetangenteenunpunto,lacurvanopuededarunsaltoenesepunto.Laformulacinprecisadeestehechoesunteoremaimportante.

Teorema.Siexiste(a),entoncesescontinuaena.

Unafuncinesderivableenunintervaloabierto(a,b)siloesentodoslosnmeroscde(a, b). Tambin se considerarn funciones que son derivables en un intervalo infinito(,a),(a,)obien( , ).

Paraintervaloscerradosusaremoslasiguientedefinicin.




Definicin. Una funcin es derivable en un intervalo cerrado [a , b], si lo es en el

intervalo(a,b)yloslmites0

( ) ( )limh

f a h f ah+

+

0

( ) ( )limh

f a h f ah

+ existen.

Los lmitespor laderechaypor la izquierdaen ladefinicinanterior,se llamanderivadaporladerechayderivadaporlaizquierdadeenayb,respectivamente.

Laderivadadeunafuncinenintervalosdelaforma[a,b),[a,),(a,b]obien(,b]sedefineusandoloslmitesporladerechaoporlaizquierdaenunodelospuntosextremos.Siestdefinidaenunintervaloabiertoquecontieneaa,entonces(a)existesiyslosilasderivadasporladerechayporlaizquierdaenaexistenysoniguales.

El inversodelprimer teorema.Es falso.Siuna funcin es continuaen c,no sesiguequetengaderivadaenc.Estoseveconfacilidadexaminandolafuncin(x)=|x|enelorigen. Esta funcin,por cierto,es continuaen cero,perono tienederivada ah.(Demostracinacargodellector)

Elargumentorecinpresentadodemuestraqueencualquierpuntoenelque lagrfica de tiene un pico o presenta una esquina aguda, es continua, pero nodiferenciable.

Suponiendo que la funcin es derivable en a, se puede enunciar la siguientedefinicin.

Definicin.Rectatangente:Lapendientedelarectatangentealagrficadeenelpunto(a,(a))es(a).

Derivadacomounafuncin

Si es derivable para todo x en un intervalo entonces, asociando a cada x elnmero (x),seobtieneunafuncin llamadaderivadade f. El valorde,enxest

dadoporelsiguientelmite.0

( ) limx

f(x x ) f(x)f xx

+ = ,(Lmiteunilateral),ntesequeelnmeroxesfijo,peroarbitrarioyellmitesetomahaciendotender x acero.Derivar(x)oencontrarladerivadade(x)significadeterminar(x).

Enlossiguientesejerciciossedeterminarlaprimeraderivadapordefinicin,olapendientedelarectatangentealacurvaenunpuntodado.




EjerciciosResueltos:

( ) 0 0

0

0

0

( ) 5

5 5( ) ( )( ) ( )

5 0( ) 0

5 ( ) .

5( ) 5 ( )

(

1) 5

)

x x

x

x

x

f x x

x x xf x x f xf x Lm f x Lmx x

x x xf x Lm IND

xx x x x x x

f x Lmx x x x

x x x xf x Lmx x x x x x

y x

x x

f x

=+ + = =

+ = = + + + = + +

+ = = + + + +

=

= 0

1 55 ( ) 2x

Lm f xx x x x

= =+ +

( ) ( ) ( )( ) ( )

2 2 2 2 22

0 0

2

0

0 0

2 ( ) 2 ( )0lim lim li

2

m0

2lim lim 2 2 ( ) 2

) ( )

x x x

x x

x x x x x x x xx x xind

x x xx x x

x x x f x

f x

x

x

x

+ + + + = = = = += = + = =

=

( ) ( )

0

0

0 0

0

5 5 0( ) lim0

5 5 5 5( ) lim

5 5

5 5 5 5

3) (

( ) lim ( ) lim5 5 5 5

1( ) l

5

im

)

x

x

x x

x

x x xf x ind

xx x x x x x

f xx x x x

x x x x x xf x f xx x x x x x x

x

x

f xx

x

x

f

+ + + = = + + + + + + + = + + + +

+ + + + + = = + + + + + + + + = +

= +

1( )2 55 5

f xxx

= ++ + +




3 3

0

3 2 2 3 3

0

2 2 3 2 2

0 0

3

2

0

( ) 4( ) ( 4 ) 0( )0

3 3 ( ) ( ) 4 4 4( )

3 3 ( ) ( ) 4 (3 3 ( ) ( ) 4)( )

4) ( )

)

(3

4

(

( )

x

x

x x

x

x x x x x xf x Lm indx

x x x x x x x x x xf x Lmx

x x x x x x x x x x xf x Lm f x Lmx

f

xf x Lm

x x

x

x

+ + = =+ + + + = + + + + = =

=

=

+ 2 23 ( ) ( ) 4) ( ) 3 4x x x f x x + =

( )( )

( )( )

( )[ ] ( ) ( )( ) ( )

( ) ( )

0

0

2 2

0

2

0

2 2 3 2 3 0( )3 3 4 3 4 0

3 4 2 2 3 3 3 4 2 3( )

3 3 4 3 4

6 6 9 8 8 12 6 9 6 9 8 12( )3 3 4 3 4

6 6 9(

2 35) ( )3

)

4

x

x

x

x

x x xf x Lm ind

x x x

x x x x x xf x Lm

x x x

x x x x x x x x x x x xf x Lmx x x

x x xf x

xx

x

Lm

f

=+ = =+ + ++ + + + = + + ++ + + + + + = + + ++

+

= ( ) ( )( ) ( ) ( ) ( )

( )

2

0 0

2

8 8 12 6 9 6 9 8 123 3 4 3 4

17 17( ) ( )3 3 4 3 4 3 3 4 3 4

17( )3 4

x x

x x x x x x x x xx x x x

xf x Lm f x Lmx x x x x x x

f xx

+ + + + + + + +

= = + + + + + + = +

2 2

0

2 2 2 2

0 0

0 0

2

3 2( ) (3 2 ) 0( ) lim0

3 2 4 2( ) 3 2 4 2( )(

6) ( )

) lim ( ) lim

( 4 2 )( ) lim ( ) lim( 4 2 ) ( )

3 2

4

x

x x

x x

x x xf xx

x x x x x x x xf x f xx x

x x

f x x

xf x f x x x f x xx

+ = = + = =

= =

=




0

1 103 ( ) 3( ) lim0

17) ( )3

x

f xx

x x xf xx

+ = =

=

0 0

0

20

(3 ) (3 ) 3 3( ) lim ( ) lim(3 )(3 ) (3 )(3 )

( ) lim(3 )(3 )

1 1( ) lim ( )(3 )(3 ) (3 )

x x

x

x

x x x x x xf x f xx x x x x x x x

xf xx x x x

f x f xx x x x

+ + = = =

= =

0 0

0 0

0

1 10 2 1 (2 2 1)2( ) 1 2 1( ) lim ( ) lim0 (2 2 1)(2 1)

2 1 2 2 1) 2( ) lim ( ) lim(2 2 1)(2 1) (2 2

18) ( )

1)(2 1)2

2 1

( ) lim(2 2 1)(

x x

x x

x

f xx

x x xx x xf x f xx x x x x

x x x xf x f xx x x x x x x x

f xx x

+

= +

+ ++ + + = = = + + ++ = = + + + + + +

= + + 22( )

2 1) (2 1)f x

x x =+ +

( )( )( ) ( ) ( )( )

( )

0

0 0

0 0

2( ) 1 2 1 0( ) lim0

2 2 1 2 1 2 2 1 2 1 2 2 1 2 1( ) lim ( ) lim

2 2 1 2 1 2 2 1 2 1

2 2 1 2 1 2( ) lim ( ) lim2 2

9) ( ) 2

2

1

1 1

x

x x

x x

x x xf x ind

xx x x x x x x x x

f x f xx x x x x x x x

x x x x

f x x

f x f xx x x x

+ + + = =+ + + + + + + + + + = = + + + + + + + +

+ + = = + + +

+

+

=

( )( ) ( )0

2 2 1 2 1

2 2 1( ) lim ( ) ( )2 12 2 1 2 1 2 1 2 1x

x x x x

f x f x f xxx x x x x

+ + + + = = = ++ + + + + + +




( )( )( )( )

( ) ( )( )( )

0

0 0

0

0

1 1 1 0( ) lim01 1

1 1 1 11 1( ) lim ( ) lim1 1 1 1 1 1

1 1( ) lim

1 1 1 1

1

( ) lim

10) ( )1

x

x x

x

x

f x indx x x x

x x x x x xx x xf x f xx x x x x x x x x x x

x x xf x

x x x x x x x

f xx

xf x

= = + + + + + + + + + ++ + + = = + + + + + + + + + +

+ + +

= +

= + + + + + + ++ = ( )( )

( )( )( )( )( )( )( ) ( ) ( ) ( ) ( )

0

0

0

20 3

1 11 1 1 1

( ) lim1 1 1 1

1( ) lim1 1 1 1

1( ) lim1 1 1 1

1 1 1( ) lim ( ) ( )1 2 1 2 ( 1)1 2 1

x

x

x

x

x xx x x x x x x

xf xx x x x x x x

f xx x x x x x

f xx x x x

f x f x f xx x xx x

+ + + + + + +

= + + + + + + + = + + + + + + +

= + + + + + = = =+ + ++ +

( ) ( ) ( ) ( )( ) ( ) ( )

( )( ) ( )( ) ( ) ( )

3 3

0

2 23 3 333 3

0 2 23 333

3 333

0 2 23 333

3

2 2 0( )0

2 2 2 2 2 2( )

2 2 2 2

22( )

11) ( ) 2

2 2 2 2

x

x

x

x x xf x Lm ind

x

x x x x x x x x xf x Lm

x x x x x x x

xx xf x Lm

x x x x x x

f x

x

x

+ = = + + + + + = + + + +

+ = +

=

+ + +




( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

0 2 23 333

0 2 23 333

22 2 33 333

2 2( )2 2 2 2

1( )2 2 2 2

1 1( ) ( )3 22 2 ( 2) 2

x

x

x x xf x Lmx x x x x x x

f x Lmx x x x x x

f x f xxx x x x

+ + = + + + + = + + + + = = + +

( )

0

0 0

0 0

1 11 0( ) ( ) ( ) 0

( ) ( ) ( )

1

12) (

( ) ( ) (

)

3

)

x

x x

x x

x x xxf x f x f x Lm indxx x x

x x xx x x x x x x x x

f x Lm f x Lmx x x xx x x x

x x xf x Lm f x Lmx x x x x xx x x

f

x x

x

x

x

x

x

+= = = =

+ + + + + = = + ++

= = + + ++

=

+ + 3 3 3

1 1 1( ) ( ) ( )2

f x f x f xx x x x x x x

= = =+ +

( ) ( ) ( )( )

0

2 2

0

1 1 0( )

113) ( )

0

( )

x

x

f x Lm x x x indx x x

x x x x x x x x xf x L

x

x

x

mx

f x

= + + = + + + + += +

= +




( )( )

( )

( ) ( )

2 2 3 2

0

3 2 2 3 2

0

2 2

0 0

2

2 2

2 ( )

2 ( )

2 1 ( ) ( )

1 1( ) ( ) 1

x

x

x x

x x x x x x x x x x xf x Lm

x x x

x x x x x x x x x x xf x Lmx x x

x x xx x x x xf x Lm f x Lmx x x x x x x x

xf x f xx x

+ + + = ++ + = +

+ + = =+ + = =

0

0

0

0

2 ( ) 2 1 1 0( )

02( ) 2 2( ) 2

1 1 1 1( )

2( ) 2 1 1

2( ) 2 1 1( )

2( ) 2

1 1

2(

( )

214) ( )1

x

x

x

x

x x xx x x

f x L m indx

x x x x x xx x x x x x

f x L mx x x x

x x xx x x

x x xf x L mx x xx

x x x

x

f x L

xfx

m

x

+ + = =+ + ++ + = + ++

+ + = + + +

=

=

( )( )

) ( 1) 2 1 1 ( 1)

2( ) 2

1 1

x x x x xx x x

x x xxx x x

+ + +

+ + +




( )2 2

0

0

0

2 2 2 2 2 2 2 1 ( 1)

( )2( ) 2

1 1

2( 1) ( 1)( )2( ) 2

1 1

2( )2( ) 2

( 1)( 1) 1 1

( )

x

x

x

x x x x x x x x xx x x

f x Lmx x xx

x x x

xx x xf x Lm

x x xxx x x

xf x Lmx x xx x x x

x x x

f x

+ ++ = + + +

+ = + + +

/ / = +/ + + +

( ) ( )

( ) ( )

0

0 02 2

4 3

22( ) 2( 1) ( 1)

1 1

2 2( ) ( )2 2 21 2 11 1 1

1 1( ) ( )2 1 (2 ) 1

1

x

x x

Lmx x xx x x

x x x

f x Lm f x Lmx x xx x

x x x

f x f xx x x xx

= ++ + + = = +

= =

( )( )( ) ( )( )

( )( )( ) ( )

( )( )( )

0

2 2

0 0

2 2

0

1 0( ) lim1 2 1 2 0

2 2 2 21 2 1 2 21( ) lim lim1 2 2 1 2 1 2 2 1 2

2 2 2

15) ( )1 2

2( ) lim1 2 2 1

x

x x

x

x x xf x indx x x x

x x x x x x x x xx x x x x xf x

x x x x x x x x

x x x x x x x x xf xx x

xf xx

x

+ = = + + + = =

+ + + =

=

( )2x




( )( ) ( )( ) ( )20 01 1( ) lim ( ) lim ( )

1 2 2 1 2 1 2 2 1 2 1 2x xxf x f x f x

x x x x x x x x = = =

( )3 30

3 2 2 3 3

0

2 22 2 3

0 0

2 2

0

3

4( ) 4 0( )0

3 3 4 4 4( )

3 3 43 3 4( ) ( )

( ) 3 3 4

16) ( ) 4

x

x

x x

x

x x x x x xf x Lm ind

xx x x x x x x x x xf x Lm

xx x x x xx x x x x xf x Lm f x Lm

x xf x L

f

m x x

x x

x

x

x

+ + + = =+ + + + =

/ + + + + = = / / = + +

=

2( ) 3 4f x x =

( ) ( ) ( )

( )

3 32 2

0

3 2 2 3 32 2 2

0

32

4 42 23 3 3 3 0( )

0

( 3 3 ( )

417) ( ) 23 3

) 2 ( ) 2 2 23 3( )

x

x

x x xx x x x x x

f x Lm Indx

x x x x x x xx x x x x x x xf x L

xf x x

x

mx

+ + + + + + = =

+ + + + + + + +

= +

=

+

3 3 32 2 2 2

0

2 2 2

0

22

0

( ) 2( ) 2 ( ) 2 2 23 3 3( )

( ) 2 ( ) 2( )

2 2( ) ( ) 2 2

x

x

x

x x xx x x x x x x x x x x xf x Lm

xx x x x x x x xf x Lm

xx x x x x x

f x Lm f x x xx

+ + + + + + =

+ + = / + +/ = = +/




( )( ) ( ) ( )

2 2

0 0 0

2

0

1 1 20 2( ) lim ( ) lim ( ) lim (

1 118) ( )

) i0

2lm 2x x x x

x x xx xf x ind f x f x f x xx x

xx

x

fx

+

+

=++ = = = = = + =

( )

( ) [ ]

0

0

0

0

0

( ) 0( ) 0

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )( )

( ) (

19)

) ( )

( ) ( )

1( )

( )

x

x

x

x

x

f x SenSen x x Sen xf x Lm ind

xCos x Sen x Sen x Cos x Sen x

f x Lmx

Cos x Sen x Sen x Cos x Sen xf x Lmx

Cos x Sen x Sen x Cos xf x Lm

xCof x L

x

m

+ = =+ =+ =

+ =

=

=

[ ]

[ ][ ]

[ ]

0

0 0

0 0

0 0

( ) ( ) 1( ) ( )

( ) 1( )( )

1 ( )( ): 1 ; 0

1 ( )( )( ) ( )

x

x x

x x

x x

Sen x Cos xs x Sen x Lmx x

Cos xSen xf x Cos x Lm Sen x Lmx x

Cos xSen xPero conocemos Lm Lmx x

Cos xSen xf x Cos x Lm Sen x Lm f x Cos xx x

+ = +

= = = =

[ ]

0

0

0

0

( ) 0( )0

( ) ( )( )

( ) ( )( )

( ) 1 ( )( )

20) ( )

x

x

x

x

Cos x x Cos xf x Lm indx

Cos xCos x Sen x Sen x Cos xf x Lmx

Cos xCos x Cos x Sen x Sen xf x Lmx

Cos x Cos x Sen x Sen xf x Lm

f x Cos x

x

+ = = = =

=

=




[ ][ ]

[ ][ ]

0 0

0 0

0 0

0 0

( ) 1 ( )( )

( ) 1 ( )( )

1 ( )( )

1 ( ): 0 ; 1 ( )

x x

x x

x x

x x

Cos x Cos x Sen x Sen xf x Lm Lmx x

Cos x Sen xf x Cos x Lm Sen xLmx xCos x Sen xf x Cos x Lm Sen xLmx x

Cos x Sen xConociendo Lm Lm f xx x

= =

= = = =

0 0: 0 1

x x

Sen x

Nota Lm Sen x LmCos x

= =

[ ]( )

0

0 0

0

( ) ( ) 0( ) ; : ( )0

21) (

1( ( )) 1 ( )

( )1 ( )1( ) ( )

(

)

( ))

x

x x

x

tg x x tg x tga tgbf x Lm ind identidad tg a bx tga tgb

tg x tg x tg x tg x tg xtg x tg x tg x tg x tg xtg xtg xf x Lm f x Lmx x

tg x tg x tgf x Lm

f x tg x

+ + = = + = + + = =

+ / / =

=

[ ] [ ]02( ) 12 ( ) ( )

1 ( ) 1 ( )x

tg x tg xx tg x tg x f x Lmx tg x tg x x tg x tg x

+ =

[ ]2 02

0 0

2 2

0 0

2

( )( ) 1 ( )1 ( ) ( )

( ) 1( ) 1 ( )1 ( ) ( )

( )( )( )( ) 1 ( ) ( ) 1 ( )( )

( ) 1 ( )

x

x x

x x

tg xf x tg x Lmx tg x tg x

tg xf x tg x Lm Lmx tg x tg xSen x

Sen xCos xf x tg x Lm f x tg x Lmx xCos x

f x tg x

= =

= =

= 20 02 2 2

( ) 1 ( ) 1 ( )

1 ( ) ( )

x x

Sen xLm Lm f x tg xx Cos x

Sec x tg x f x Sec x

=

= =




( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

0

0 0 0

2 2 0( ) lim0

2 2 2 2 1 2 1( ) lim ( ) 2 lim ; :

22) ( ) 2

lim 1

( ) 2 (1) ( ) 2

x x x

x

x x x x xx

x x

x x

x

x

f x indx

f x f x perox x x

f x f x

f x+

= = = = =

= =

=

0 0

0 0

0( ) ( )

23) ( )

01 1( ) : 1 ( )

x x x x x x

x x

x xx x

x x

x

e e e e ef x Lm ind f x Lmx x

e ef x Lm

f x e

Pero Lm f xe ex x

+

= = = = = =

=

0

24) ( ) ( )

( ) 0( )0

a a

x

x x xLog Logf x indL

f x xLog

m x

a

+ = =

=

[ ] ( )( )

00

1

0

111

0 0

0

1 ( ) 1( ) ( ) 1 ; 0

( ) 1 : ; 0 0

( ) ( )1 1

1( ) 1

a xx

x

x

xtxtt t

t

x x xf x f x Lm xLog LogLm ax x x x

x xf x Lm Cambio t x tx si x tLoga x x

f x Lm f x LmtLog Loga a t

f x LmLoga x

+ = = + > = + = = = =

= =+ + = +

( ) ( )( )

11

0

1

0

1( ) 1

1: 1 ( ) ( )

ttt

tat

f x Lm tLogt ax

Pero Lm t e f x Log ex

= +

+ = =




( )

0

0 0 0

1

0 0

( ) ( ) 0( )0

( ) ( ) 1( ) ( ) ( )

1

25) ( ) ( )

( ) (1 ) ( ) 1

x

x x x

x

x x

Lna x x Ln axf x Lmx

a x x x x xLn Ln Lnax x xf x Lm f x Lm f x Lmx x x

x xf x Lm Ln f x Lm

f x

Ln

L a

x x x

n x

+ = =+ + + = = =

= + = +

=

( ) ( ) ( )( )

1 1 1

0 0 0

1

0

0 0

1 1( ) ( ) ( )1 1 1

1 1( ) ( ) ( )1

tx t tt t t

tt

xCambio t x tx x tx

f x LmLn f x Lm Ln f x LnLmt t tx x

Pero Lm e f x Ln e f xt x x

= =

= = =+ + +

= = =+

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