Esercizi svolti

 

Trovare le equazioni degli asintoti della seguente funzione:

f(x)= x 2 x4 x1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaiabg2da9maalaaabaGaamiEamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaadIhacqGHsislcaaI0aaabaGaamiEaiabgk HiTiaaigdaaaaaaa@4277@

La funzione assegnata è definita nel seguente insieme di esistenza:

D=] ,1 [] 1,+ [ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maajmcabaGaaGPaVlaaykW7cqGHsislcaaMc8UaeyOhIuQaaiil aiaaykW7caaMc8UaaGymaiaaykW7caaMc8oacaGLDbGaay5waaGaey OkIG8aaKWiaeaacaaMc8UaaGPaVlaaigdacaGGSaGaaGPaVlaaykW7 cqGHRaWkcaaMc8UaeyOhIuQaaGPaVlaaykW7aiaaw2facaGLBbaaaa a@5AB6@

Calcolando i limiti della funzione nei punti x=±,x=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iabgglaXkabg6HiLkaacYcacaaMc8UaaGPaVlaaykW7caWG4bGa eyypa0JaaGymaiaaykW7caGGSaaaaa@45A2@  si ha:

lim x 1 f(x)= lim x 1 x 2 x4 x1 =+; lim x 1 + f(x)= lim x 1 + x 2 x4 x1 =; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWfqa qcfayaaiGacYgacaGGPbGaaiyBaaadbaqcLbmacaWG4bGaeyOKH4Qa aGymaSWaaWbaaWqabeaacqGHsislaaaaleqaaOGaaGPaVlaadAgaca GGOaGaamiEaiaacMcacqGH9aqpdaWfqaqcfayaaiGacYgacaGGPbGa aiyBaaadbaqcLbmacaWG4bGaeyOKH4QaaGymaSWaaWbaaWqabeaacq GHsislaaaaleqaaOGaaGPaVpaalaaabaGaamiEamaaCaaaleqabaGa aGOmaaaakiabgkHiTiaadIhacqGHsislcaaI0aaabaGaamiEaiabgk HiTiaaigdaaaGaeyypa0JaaGPaVlabgUcaRiaaykW7cqGHEisPcaaM c8Uaai4oaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7aeaadaWfqaqcfayaaiGacYgacaGGPbGa aiyBaaadbaqcLbmacaWG4bGaeyOKH4QaaGymaSWaaWbaaWqabeaacq GHRaWkaaaaleqaaOGaaGPaVlaaykW7caWGMbGaaiikaiaadIhacaGG PaGaeyypa0ZaaCbeaKqbagaaciGGSbGaaiyAaiaac2gaaWqaaKqzad GaamiEaiabgkziUkaaigdalmaaCaaameqabaGaey4kaScaaaWcbeaa kiaaykW7daWcaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsi slcaWG4bGaeyOeI0IaaGinaaqaaiaadIhacqGHsislcaaIXaaaaiab g2da9iaaykW7cqGHsislcaaMc8UaeyOhIuQaaGPaVlaacUdaaaaa@9E23@

Quindi la funzione f(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaaaa@3937@  ha in x=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2 da9iaaigdaaaa@38B4@  un asintoto verticale.

La funzione non ammette asintoti orizzontali in quanto:

lim x f(x)= lim x x 2 x4 x1 = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaKqbag aaciGGSbGaaiyAaiaac2gaaWqaaKqzadGaamiEaiabgkziUkabg6Hi LcWcbeaakiaaykW7caWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaC beaKqbagaaciGGSbGaaiyAaiaac2gaaWqaaKqzadGaamiEaiabgkzi Ukabg6HiLcWcbeaakmaalaaabaGaamiEamaaCaaaleqabaGaaGOmaa aakiabgkHiTiaadIhacqGHsislcaaI0aaabaGaamiEaiabgkHiTiaa igdaaaGaeyypa0JaeyOhIukaaa@58E5@

Questo risultato implica però la possibilità che esistano asintoti obliqui per la cui equazione, y=mx+q MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 da9iaad2gacaaMc8UaamiEaiabgUcaRiaadghaaaa@3D4C@ ,  occorre determinare:

m= lim x f(x) x = lim x x 2 x4 x 2 x =1, q= lim x [ f(x)mx ]= lim x 4 x1 =0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGTb Gaeyypa0ZaaCbeaKqbagaaciGGSbGaaiyAaiaac2gaaWqaaKqzadGa amiEaiabgkziUkabg6HiLcWcbeaakmaalaaabaGaamOzaiaacIcaca WG4bGaaiykaaqaaiaadIhaaaGaeyypa0ZaaCbeaKqbagaaciGGSbGa aiyAaiaac2gaaWqaaKqzadGaamiEaiabgkziUkabg6HiLcWcbeaakm aalaaabaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadIha cqGHsislcaaI0aaabaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgk HiTiaadIhaaaGaeyypa0JaaGymaiaaykW7caGGSaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVd qaaiaadghacqGH9aqpdaWfqaqcfayaaiGacYgacaGGPbGaaiyBaaad baqcLbmacaWG4bGaeyOKH4QaeyOhIukaleqaaOWaamWaaeaacaaMc8 UaaGPaVlaadAgacaGGOaGaamiEaiaacMcacqGHsislcaWGTbGaaGPa VlaadIhacaaMc8UaaGPaVdGaay5waiaaw2faaiabg2da9maaxabaju aGbaGaciiBaiaacMgacaGGTbaameaajugWaiaadIhacqGHsgIRcqGH EisPaSqabaGccqGHsisldaWcaaqaaiaaisdaaeaacaWG4bGaeyOeI0 IaaGymaaaacqGH9aqpcaaIWaGaaGPaVlaac6caaaaa@9A1F@

Pertanto y=x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2 da9iaadIhaaaa@38F7@  è asintoto obliquo destro e sinistro per la funzione.