Print this chapterPrint this chapter

Resum Límits i continuïtat

Cassos possibles en el càlcul de límits

Indeterminació ∞ - ∞

Com resolc la indeterminació ∞ - ∞ ?

Aquest tipus d'indeterminacions apareix quan es calcula el límit en l'infinit de diferents funcions polinòmiques, racionals o irracionals. Per a cada cas hi ha una tècnica diferent per resoldre-la

CAS A: límit de polinomis

En aquest cas el límit del polinomi queda igual que el límit dels terme de major grau

space box enclose space limit as x rightwards arrow infinity of 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 0 equals limit as x rightwards arrow infinity of a subscript n x to the power of n space space end enclose

CAS B: quan tenim una suma o resta de fraccions algebraiques

En aquest cas cal reduir l'expressió a una única fracció algebraica (fent la suma o resta de fraccions i tornar a fer el límit. Normalment torna a sortir una indeterminació però més fàcil de resoldre (normalment infinity over infinity)

CAS B: quan tenim una suma o resta d'una funció irracional amb un polinomi o una fracció algebraica

En aquest cas cal multiplicar tota l'expressió pel conjugat d'ella mateixa, arreglar i tornar a fer el límit. Normalment torna a sortir una indeterminació però més fàcil de resoldre (normalment infinity over infinity)

 

Exemple 1 (CAS A):

stack l i m with x rightwards arrow plus infinity below 3 x squared minus 5 x cubed equals limit as x rightwards arrow plus infinity of minus 5 x cubed equals negative 5 times left parenthesis plus infinity right parenthesis equals space box enclose space minus infinity space end enclose space

Exemple 2 (CAS B):

stack l i m with x rightwards arrow negative infinity below open parentheses fraction numerator x squared plus 1 over denominator space space x end fraction plus fraction numerator 5 x minus x squared over denominator 3 x minus 1 end fraction close parentheses equals open parentheses fraction numerator plus infinity over denominator negative infinity end fraction i n d e t plus fraction numerator negative infinity over denominator negative infinity end fraction i n d e t close parentheses space space equals open parentheses negative infinity plus infinity close parentheses space i n d e t space equals space space space space space space space space space space space space space space space space limit as x rightwards arrow negative infinity of open parentheses fraction numerator left parenthesis x squared plus 1 right parenthesis left parenthesis 3 x minus 1 right parenthesis over denominator space space x left parenthesis 3 x minus 1 right parenthesis end fraction plus fraction numerator x left parenthesis 5 x minus x squared right parenthesis over denominator x left parenthesis 3 x minus 1 right parenthesis end fraction close parentheses equals limit as x rightwards arrow negative infinity of open parentheses fraction numerator 3 x cubed minus x squared plus 3 x minus 1 over denominator space space x left parenthesis 3 x minus 1 right parenthesis end fraction plus fraction numerator 5 x squared minus x cubed over denominator x left parenthesis 3 x minus 1 right parenthesis end fraction close parentheses equals space space space space space space space space space space space space space space space space limit as x rightwards arrow negative infinity of open parentheses fraction numerator 2 x cubed plus 4 x squared plus 3 x minus 1 over denominator space space 3 x squared minus x end fraction close parentheses equals fraction numerator negative infinity over denominator plus infinity end fraction i n d e t space equals space box enclose space minus infinity space end enclose space space

Exemple 3 (CAS C):

stack l i m with x rightwards arrow plus infinity below open parentheses x minus square root of x squared minus 1 end root close parentheses equals plus infinity minus infinity space i n d e t space equals limit as x rightwards arrow plus infinity of fraction numerator open parentheses x minus square root of x squared minus 1 end root close parentheses open parentheses bold x bold plus square root of bold x to the power of bold 2 bold minus bold 1 end root close parentheses over denominator open parentheses bold x bold plus square root of bold x to the power of bold 2 bold minus bold 1 end root close parentheses end fraction equals limit as x rightwards arrow plus infinity of fraction numerator x squared minus open parentheses square root of x squared minus 1 end root close parentheses squared over denominator straight x plus square root of straight x squared minus 1 end root end fraction equals limit as x rightwards arrow plus infinity of space space space space space space space space space space space space space space equals limit as x rightwards arrow plus infinity of fraction numerator x squared minus left parenthesis x squared minus 1 right parenthesis over denominator straight x plus square root of straight x squared minus 1 end root end fraction equals limit as x rightwards arrow plus infinity of fraction numerator 1 over denominator straight x plus square root of straight x squared minus 1 end root end fraction equals fraction numerator 1 over denominator plus infinity plus infinity end fraction equals fraction numerator 1 over denominator plus infinity end fraction equals space space box enclose space 0 space end enclose