Previous
Next

LÍMITS. Concepte

Resolució indeterminació 0/0

Com resolc la indeterminació 0/0?

Dependrà de l'expressió de f(x). Normalment ens podem trobar amb dos casos:

CAS A:

f(x) és una funció racional, és a dir una divisió entre dos polinomis. En aquest cas procedim a factoritzar els dos polinomis, simplificar i finalment tornar a fer el límit

CAS B:

L'expressió de f(x) és una fracció on el numerador o el denominador ( i sols 1 d'ells) és un polinomi i l'altre ( el numerador o el denominador) és una suma o resta on la x està sota una arrel quadrada.

En aquest cas procedim a multiplicar el numerador i denominador pel conjugat de l'expressió que conté l'arrel, factoritzar els polinomis, simplificar i finalment tornar a fer el límit .

Exemple 1 (CAS A):

limit as x rightwards arrow 7 of fraction numerator 7 x minus x squared over denominator 49 minus x squared end fraction equals fraction numerator 49 minus 49 over denominator 49 minus 49 end fraction equals 0 over 0 space i n d e t space equals limit as x rightwards arrow 7 of fraction numerator x times left parenthesis 7 minus x right parenthesis over denominator left parenthesis 7 minus x right parenthesis times left parenthesis 7 plus x right parenthesis end fraction equals limit as x rightwards arrow 7 of fraction numerator x times up diagonal strike left parenthesis 7 minus x right parenthesis end strike over denominator up diagonal strike left parenthesis 7 minus x right parenthesis end strike times left parenthesis 7 plus x right parenthesis end fraction equals space space space space space space space space space space space space space space space space space space space space space limit as x rightwards arrow 7 of fraction numerator x over denominator 7 plus x end fraction equals 7 over 14 equals space box enclose 1 half end enclose

Exemple 2 (CAS A):

limit as x rightwards arrow 1 of fraction numerator x squared minus 2 x plus 1 over denominator space space 2 x squared plus 4 x minus 6 end fraction equals fraction numerator 1 minus 2 plus 1 over denominator 2 plus 4 minus 6 end fraction equals 0 over 0 space i n d e t space equals limit as x rightwards arrow 1 of fraction numerator left parenthesis x minus 1 right parenthesis squared over denominator 2 times left parenthesis x minus 1 right parenthesis times left parenthesis x plus 3 right parenthesis end fraction equals limit as x rightwards arrow 1 of fraction numerator up diagonal strike left parenthesis x minus 1 right parenthesis end strike times left parenthesis x minus 1 right parenthesis over denominator 2 times up diagonal strike left parenthesis x minus 1 right parenthesis end strike times left parenthesis x plus 3 right parenthesis end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as x rightwards arrow 1 of fraction numerator x minus 1 over denominator 2 x plus 6 end fraction equals 0 over 8 equals space box enclose 0

Exemple 3 (CAS B):

limit as space space space space space x rightwards arrow negative 2 of space fraction numerator square root of 6 plus x end root plus x over denominator space space x plus 2 end fraction equals fraction numerator square root of 4 minus 2 over denominator negative 2 plus 2 end fraction equals 0 over 0 space i n d e t space equals limit as x rightwards arrow negative 2 of space fraction numerator open parentheses square root of 6 plus x end root plus x close parentheses open parentheses square root of 6 plus x end root minus x close parentheses over denominator space open parentheses space x plus 2 close parentheses open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as space space x rightwards arrow negative 2 of space fraction numerator open parentheses square root of 6 plus x end root close parentheses squared minus x squared over denominator space open parentheses space x plus 2 close parentheses open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals limit as space space space space space x rightwards arrow negative 2 of space fraction numerator 6 plus x minus x squared over denominator space open parentheses space x plus 2 close parentheses open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as space space space space space x rightwards arrow negative 2 of space fraction numerator negative left parenthesis x squared minus x minus 6 right parenthesis over denominator space open parentheses space x plus 2 close parentheses open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals limit as space space space space space x rightwards arrow negative 2 of space fraction numerator negative up diagonal strike left parenthesis x plus 2 right parenthesis space end strike left parenthesis x minus 3 right parenthesis over denominator space up diagonal strike open parentheses space x plus 2 close parentheses end strike space open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as space space space space x rightwards arrow negative 2 of space fraction numerator negative x plus 3 over denominator space space open parentheses square root of 6 plus x end root minus x close parentheses end fraction equals fraction numerator 5 over denominator square root of 4 plus 2 end fraction equals box enclose space 5 over 4 space end enclose

Exemple 4 (CAS B):

limit as x rightwards arrow 7 of space space fraction numerator 7 x minus x squared over denominator 3 minus square root of x plus 2 end root end fraction equals fraction numerator 49 minus 49 over denominator 3 minus square root of 9 end fraction equals 0 over 0 space i n d e t space equals limit as x rightwards arrow 7 of fraction numerator open parentheses 7 x minus x squared close parentheses times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator left parenthesis 3 minus square root of x plus 2 end root right parenthesis times left parenthesis 3 plus square root of x plus 2 end root right parenthesis end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space limit as x rightwards arrow 7 of fraction numerator open parentheses 7 x minus x squared close parentheses times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator 9 minus open parentheses square root of x plus 2 end root close parentheses squared end fraction equals limit as x rightwards arrow 7 of fraction numerator open parentheses 7 x minus x squared close parentheses times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator 9 minus x minus 2 end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as x rightwards arrow 7 of fraction numerator open parentheses 7 x minus x squared close parentheses times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator 7 minus x end fraction equals limit as x rightwards arrow 7 of fraction numerator x times up diagonal strike open parentheses 7 minus x close parentheses end strike times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator up diagonal strike left parenthesis 7 minus x right parenthesis end strike end fraction equals space space space space space space space space space space space space space space space space space space space space space space space space space space limit as x rightwards arrow 7 of fraction numerator x times left parenthesis 3 plus square root of x plus 2 end root right parenthesis over denominator 1 end fraction equals limit as x rightwards arrow 7 of space x times open parentheses 3 plus square root of x plus 2 end root close parentheses equals 7 times left parenthesis 3 plus 3 right parenthesis equals space box enclose space 42 space end enclose

Previous
Next