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LÍMITS

És el mateix límit que imatge?

No, no és el mateix.

Majoritàriament el límit coincideix amb la imatge però no sempre.

Exemple 1:

Observa la funció f left parenthesis x right parenthesis equals fraction numerator x minus 1 over denominator x squared minus 1 end fraction

Si de bon començament intentem trobar la imatge per x=1 veiem que no existeix ja que no es pot dividir per zero.

f left parenthesis 1 right parenthesis equals fraction numerator 1 minus 1 over denominator 1 squared minus 1 end fraction equals 0 over 0 space q u e space n o space e x i s t e i x space rightwards arrow x equals 1 space n o space é s space d e l space d o m i n i space d e space l a space f u n c i ó

Però en canvi si que existeix f(0'9), f(0'99), f(0'999)... i també f(1'1), f(1'01), f(1'001)...imatges pels valors de x propers a x=1

Aleshores la idea de límit és el nombre al que s'acosten les imatges quan x s'acosta al nombre on fem el límit

f(0'9)= 0'526315

f(0'99)= 0'502512

f(0'999)= 0'50025

f(0'9999)= 0'500025

open table attributes columnalign right end attributes row cell open table attributes columnalign right end attributes row cell f left parenthesis 0 apostrophe 9 right parenthesis equals space 0 apostrophe 526315 space space f left parenthesis 0 apostrophe 99 right parenthesis equals space 0 apostrophe 502512 space space f left parenthesis 0 apostrophe 999 right parenthesis equals space 0 apostrophe 50025 space space f left parenthesis 0 apostrophe 9999 right parenthesis equals space 0 apostrophe 500025 end cell row cell... end cell end table close curly brackets rightwards arrow limit as x rightwards arrow 1 to the power of plus of f left parenthesis x right parenthesis equals 0 apostrophe 5 open table attributes columnalign right end attributes row cell f left parenthesis 1 apostrophe 1 right parenthesis equals space 0 apostrophe 476149 space space f left parenthesis 1 apostrophe 01 right parenthesis equals space 0 apostrophe 497512 space space f left parenthesis 1 apostrophe 001 right parenthesis equals space 0 apostrophe 49975 space space f left parenthesis 1 apostrophe 0001 right parenthesis equals space 0 apostrophe 499975 end cell row cell... end cell end table close curly brackets rightwards arrow limit as x rightwards arrow 1 to the power of minus of f left parenthesis x right parenthesis equals 0 apostrophe 5 end cell row blank end table close curly brackets space space space space space rightwards double arrow limit as x rightwards arrow 1 of fraction numerator x minus 1 over denominator x squared minus 1 end fraction equals 0 apostrophe 5

Exemple 2:

Sigui la funció

f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 3 x minus 7 space space s i space space space x less than 2 end cell row cell x plus 3 space space space space s i space space space x greater or equal than 2 end cell end table close E n space a q u e s t space c a s space t e n i m space f left parenthesis 2 right parenthesis equals 2 plus 3 equals 5 P e r ò space e l space l í m i t space e n space x equals 2 space n o space e x i s t e i x space j a space q u e space e l s space l í m i t s space l a t e r a l s space s ó n space d i f e r e n t s colon open table attributes columnalign right end attributes row cell limit as x rightwards arrow 2 to the power of minus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 2 to the power of minus of 3 x minus 7 equals 6 minus 7 equals negative 1 end cell row cell limit as x rightwards arrow 2 to the power of plus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 2 to the power of plus of x plus 3 equals 2 plus 3 equals 5 end cell end table close curly brackets rightwards arrow limit as x rightwards arrow 2 of f left parenthesis x right parenthesis space n o space e x i s t e i x

Exemple 3: Donada la funció f(x)=3x-4 en aquest cas el límit coincideix amb la imatge per a qualsevol valor de la x

Exemple 4: Sigui la funció

f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell 3 x minus 1 space space s i space space space x less than 2 end cell row cell x plus 3 space space space space s i space space space x greater or equal than 2 end cell end table close E n space a q u e s t space c a s space t e n i m space f left parenthesis 2 right parenthesis equals 2 plus 3 equals 5 I space a space m é s space e l space l í m i t space e n space x equals 2 space space e x i s t e i x space j a space q u e space e l s space l í m i t s space l a t e r a l s space c o i n c i d e i x e n colon open table attributes columnalign right end attributes row cell limit as x rightwards arrow 2 to the power of minus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 2 to the power of minus of 3 x minus 1 equals 6 minus 1 equals 5 end cell row cell limit as x rightwards arrow 2 to the power of plus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 2 to the power of plus of x plus 3 equals 2 plus 3 equals 5 end cell end table close curly brackets rightwards arrow limit as x rightwards arrow 2 of f left parenthesis x right parenthesis space equals 5 E n space a q u e s t space c a s space e l space l í m i t space e n space x equals 2 space c o i n c i d e i x space a m b space l a space i m a t g e limit as x rightwards arrow 2 of f left parenthesis x right parenthesis space equals f left parenthesis 2 right parenthesis equals 5

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