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Límits de funcions racionals

Les funcions racionals són les formades per la divisió de dos polinomis f(x)= \frac{P(x)}{Q(x)}

on P left parenthesis x right parenthesis equals a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus... plus a subscript 1 x plus a subscript 0 space a m b space a subscript i space space end subscript element of R.

on Q left parenthesis x right parenthesis equals b subscript m x to the power of m plus b subscript m minus 1 end subscript x to the power of m minus 1 end exponent plus... plus b subscript 1 x plus b subscript 0 space a m b space b subscript i space space end subscript element of R.

Límits en un punt

Per fer els límits en un punt p en principi substituirem la funció en el punt.

  • Si dóna un valor finit aquest serà el límit.
  • En cas que doni k over 0  el límit serà plus infinity space ó space minus infinity en funció del signe de la k i del 0. Caldrà fer els límits laterals.
  • Si dóna 0 over 0 es tracta d'una indeterminació que caldrà resoldre. Haurem de factoritzar numerador i denominadors, simplificar i després tornar a fer el límit. Les treballarem una mica més endavant.


Exemples

  • f left parenthesis x right parenthesis equals fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction  calculem els límits en els punts 3, 5 i -1.
    • limit as x rightwards arrow 3 of f left parenthesis x right parenthesis equals limit as x rightwards arrow 3 of fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction equals fraction numerator 3 squared plus 3 times 3 plus 2 over denominator 3 squared minus 4 times 3 minus 5 end fraction equals fraction numerator 20 over denominator negative 8 end fraction equals negative 5 over 2

    • limit as x rightwards arrow 5 of f left parenthesis x right parenthesis equals limit as x rightwards arrow 5 of fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction equals fraction numerator 5 squared plus 3 times 5 plus 2 over denominator 5 squared minus 4 times 5 minus 5 end fraction equals 42 over 0  com dividim per 0, serà plus infinity space ó space minus infinity . Caldrà estudiar els límits laterals per saber-ho.


         limit as x rightwards arrow 5 to the power of minus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 5 to the power of minus of fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction equals fraction numerator left parenthesis 4 comma 99 right parenthesis squared plus 3 times left parenthesis 4 comma 99 right parenthesis plus 2 over denominator left parenthesis 4 comma 99 right parenthesis squared minus 4 times left parenthesis 4 comma 99 right parenthesis minus 5 end fraction equals 42 over 0 to the power of minus equals negative infinity

         limit as x rightwards arrow 5 to the power of plus of f left parenthesis x right parenthesis equals limit as x rightwards arrow 5 to the power of plus of fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction equals fraction numerator left parenthesis 5 comma 01 right parenthesis squared plus 3 times left parenthesis 5 comma 01 right parenthesis plus 2 over denominator left parenthesis 5 comma 01 right parenthesis squared minus 4 times left parenthesis 5 comma 01 right parenthesis minus 5 end fraction equals 42 over 0 to the power of plus equals plus infinity

         Com els dos límits laterals prenen valors diferents, el límit en 5 no existeix.

    • limit as x rightwards arrow negative 1 of f left parenthesis x right parenthesis equals limit as x rightwards arrow negative 1 of fraction numerator x squared plus 3 x plus 2 over denominator x squared minus 4 x minus 5 end fraction equals fraction numerator left parenthesis negative 1 right parenthesis squared plus 3 times left parenthesis negative 1 right parenthesis plus 2 over denominator left parenthesis negative 1 right parenthesis squared minus 4 times left parenthesis negative 1 right parenthesis minus 5 end fraction equals 0 over 0 space i n d e t e r m i n a c i ó
      Aprendrem més endavant a resoldre aquest tipus d'indeterminacions.

Límits a infinit

Per fer límits a plus infinity space ó space minus infinity només ens fixarem en el terme de grau més alt de cada polinomi i aquests termes ens donaran el límit. És a dir ens fixarem en

a subscript n x to the power of n del polinomi P(x) i en b subscript m x to the power of m del polinomi Q(x)

El límit final dependrà del grau de la següent manera:

box enclose 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 subscript n x to the power of n over denominator b subscript m x to the power of m end fraction equals open curly brackets table attributes columnalign left end attributes row cell plus infinity space o space minus infinity space s i space g r a u space n u m e r a d o r greater than g r a u space d e l space d e n o m i n a d o r space space end cell row cell 0 space space s i space space space space space space g r a u space n u m e r a d o r less than g r a u space d e l space d e n o m i n a d o r space space end cell row cell a subscript n over b subscript m space s i space g r a u space n u m e r a d o r equals g r a u space d e l space d e n o m i n a d o r space end cell end table close end enclose


El signe de l'infinit caldrà estudiar-lo amb detall seguint el que s'ha comentat amb les funcions polinòmiques i la regla dels signes.

Exemples
  • f left parenthesis x right parenthesis equals fraction numerator negative x cubed plus 2 x squared minus 5 over denominator 3 x squared plus 4 x minus 1 end fraction

limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow plus infinity of fraction numerator negative x cubed plus 2 x squared minus 5 over denominator 3 x squared plus 4 x minus 1 end fraction equals limit as x rightwards arrow plus infinity of fraction numerator negative x cubed over denominator 3 x squared end fraction equals fraction numerator g r a u 3 over denominator g r a u space 2 end fraction equals negative infinity  el signe és negatiu perquè el numerador és negatiu i denominador positiu


limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow negative infinity of fraction numerator negative x cubed plus 2 x squared minus 5 over denominator 3 x squared plus 4 x minus 1 end fraction equals limit as x rightwards arrow negative infinity of fraction numerator negative x cubed over denominator 3 x squared end fraction equals fraction numerator g r a u 3 over denominator g r a u space 2 end fraction equals plus infinity space , observem que en aquest cas, el numerador tindria signe + i el denominador també i per això el quocient és +.

  • f left parenthesis x right parenthesis equals fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x to the power of 5 plus 4 x minus 1 end fraction
limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow plus infinity of fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x to the power of 5 plus 4 x minus 1 end fraction equals limit as x rightwards arrow plus infinity of fraction numerator 2 x cubed over denominator 3 x to the power of 5 end fraction equals fraction numerator g r a u 3 over denominator g r a u space 5 end fraction equals 0
limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow negative infinity of fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x to the power of 5 plus 4 x minus 1 end fraction equals limit as x rightwards arrow negative infinity of fraction numerator 2 x cubed over denominator 3 x to the power of 5 end fraction equals fraction numerator g r a u 3 over denominator g r a u space 5 end fraction equals 0


  • f left parenthesis x right parenthesis equals fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x cubed plus 4 x minus 1 end fraction

    limit as x rightwards arrow plus infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow plus infinity of fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x cubed plus 4 x minus 1 end fraction equals limit as x rightwards arrow plus infinity of fraction numerator 2 x cubed over denominator 3 x cubed end fraction equals fraction numerator g r a u 3 over denominator g r a u space 3 end fraction equals 2 over 3
    limit as x rightwards arrow negative infinity of f left parenthesis x right parenthesis equals limit as x rightwards arrow negative infinity of fraction numerator 2 x cubed plus 2 x squared minus 5 over denominator 3 x cubed plus 4 x minus 1 end fraction equals limit as x rightwards arrow negative infinity of fraction numerator 2 x cubed over denominator 3 x cubed end fraction equals fraction numerator g r a u 3 over denominator g r a u space 3 end fraction equals 2 over 3
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