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LÍMITS. Concepte

Resolució indeterminació ∞/∞

Com resolc la indeterminació bold infinity over bold infinity?

Dependrà de l'expressió . Bàsicament ens podem trobar amb dos casos:

CAS A:

L'expressió és una divisió entre dos polinomis. En aquest cas procedim a comparar els graus dels polinomis

Imaginem que l'expressió és del tipus fraction numerator p left parenthesis x right parenthesis over denominator q left parenthesis x right parenthesis end fraction space space o n space p left parenthesis x right parenthesis space equals a x to the power of m plus... space space space space i space q left parenthesis x right parenthesis equals b x to the power of n plus... i m i n són els termes de major grau dels polinomis p(x) i q(x). Llavors:

  •  S i space space g r a u left parenthesis p left parenthesis x right parenthesis right parenthesis greater than g r a u left parenthesis q left parenthesis x right parenthesis right parenthesis space space rightwards arrow limit as x rightwards arrow infinity of fraction numerator p left parenthesis x right parenthesis over denominator q left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow infinity of fraction numerator a x to the power of m over denominator b x to the power of n end fraction equals limit as x rightwards arrow infinity of a over b x to the power of m minus n end exponent equals plus-or-minus infinity
  • S i space space g r a u left parenthesis p left parenthesis x right parenthesis right parenthesis less than g r a u left parenthesis q left parenthesis x right parenthesis right parenthesis space space rightwards arrow limit as x rightwards arrow infinity of fraction numerator p left parenthesis x right parenthesis over denominator q left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow infinity of fraction numerator a x to the power of m over denominator b x to the power of n end fraction equals 0
  • S i space space g r a u left parenthesis p left parenthesis x right parenthesis right parenthesis equals g r a u left parenthesis q left parenthesis x right parenthesis right parenthesis space space rightwards arrow limit as x rightwards arrow infinity of fraction numerator p left parenthesis x right parenthesis over denominator q left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow infinity of fraction numerator a x to the power of m over denominator b x to the power of n end fraction equals limit as x rightwards arrow infinity of a over b equals a over b

CAS B:

L'expressió és una fracció on el numerador o el denominador no tenen perquè ser polinomis. Per exemple pot aparèixer la funció exponencial o la logarítmica

En aquest cas procedim a mirar quina de les dues expressions (numerador o denominador) tendeix més ràpidament cap a infinit.

Les funcions exponencials són les que creixen més ràpidament, després van les polinòmiques i les últimes són les logarítmiques. El que fem és doncs considerar el límit de la que creix més lentament com si fòs una constant (del mateix signe que l'expressió) i tornem a fer el límit.

Exemple 1 (CAS A):

limit as x rightwards arrow plus infinity of fraction numerator 7 x minus x squared over denominator 49 plus 2 x squared end fraction equals fraction numerator negative infinity over denominator plus infinity end fraction space i n d e t space equals limit as x plus infinity of fraction numerator negative x squared over denominator 2 x squared end fraction equals limit as x plus infinity of fraction numerator negative 1 over denominator 2 end fraction equals space box enclose negative 1 half end enclose

Exemple 2 (CAS A):

limit as x rightwards arrow negative infinity of fraction numerator negative 2 x plus 1 over denominator space space x squared plus 4 x minus 6 end fraction equals fraction numerator plus infinity over denominator plus infinity end fraction space i n d e t space equals limit as x rightwards arrow negative infinity of fraction numerator negative 2 x over denominator space space x squared end fraction equals limit as x rightwards arrow negative infinity of fraction numerator negative 2 over denominator space space x end fraction equals fraction numerator negative 2 over denominator negative infinity end fraction equals space box enclose 0

Exemple 3 (CAS A):

limit as x rightwards arrow plus infinity of fraction numerator negative 2 x to the power of 4 plus 1 over denominator space 4 x squared minus 6 x end fraction equals fraction numerator negative infinity over denominator plus infinity end fraction space i n d e t space equals limit as x rightwards arrow plus infinity of fraction numerator negative 2 x to the power of 4 over denominator space 4 x squared end fraction equals limit as x rightwards arrow plus infinity of fraction numerator negative 2 x squared over denominator space 4 end fraction equals limit as x rightwards arrow plus infinity of fraction numerator negative 1 over denominator space 2 end fraction x squared equals fraction numerator negative 1 over denominator space 2 end fraction times left parenthesis plus infinity right parenthesis equals space box enclose space minus infinity space end enclose

Exemple 4 (CAS B):

limit as space space space space space x rightwards arrow plus infinity of space fraction numerator 2 x plus e to the power of x over denominator space space x plus 2 end fraction equals fraction numerator plus infinity over denominator plus infinity end fraction space i n d e t space equals limit as space space space space space x rightwards arrow plus infinity of space fraction numerator 2 x plus e to the power of x over denominator space space K end fraction equals fraction numerator plus infinity over denominator k end fraction equals space space box enclose space plus infinity space end enclose space space space left parenthesis o n space k greater than 0 right parenthesis

Exemple 5 (CAS B):

limit as space space space space space x rightwards arrow plus infinity of space fraction numerator log left parenthesis x plus 1 right parenthesis over denominator space space x plus 2 end fraction equals fraction numerator plus infinity over denominator plus infinity end fraction space i n d e t space equals limit as space space space space space x rightwards arrow plus infinity of space fraction numerator K over denominator space space x plus 2 end fraction equals fraction numerator K over denominator plus infinity end fraction equals space space box enclose space 0 space end enclose space space space left parenthesis o n space k greater than 0 right parenthesis

Exemple 6 (CAS B):

limit as space space space space space x rightwards arrow plus infinity of space fraction numerator e to the power of x plus 1 end exponent over denominator negative 2 space log left parenthesis x plus 1 right parenthesis end fraction equals fraction numerator plus infinity over denominator negative 2 left parenthesis plus infinity right parenthesis end fraction equals fraction numerator plus infinity over denominator negative infinity end fraction space i n d e t space equals limit as space space space space space x rightwards arrow plus infinity of space fraction numerator e to the power of x plus 1 end exponent over denominator space K end fraction equals fraction numerator plus infinity over denominator k end fraction equals space space box enclose space minus infinity space end enclose space space space left parenthesis o n space k less than 0 right parenthesis

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