Răspuns:
Explicație pas cu pas:
a) lim x->inf(n*x^3 +1)/((nx)^3 -1) =
lim x->inf(n*x^3 +1)/(n^3*x^3 -1) = n/n^3 = 1/n^2
(impartim sus si jos cu x^3)
b) f'(x) = (1*(x^3 -1) -(x+1)*3x^2)/(x^3 -1)^2 =
(x^3 -1 -3x^3 -3x^3)/(x^3 -1)^2 =
-(2x^3 +3x^2 +1)/(x^3 -1)^2 < 0 pt. x in [0, inf),
adica f e descr. pe [0, inf)
c) Asimpt. verticala : x = 1 (unde numitorul nu e definit)
lim x -> -+inf f(x) = 0 (grad numarator < grad numitor)
Asimpt. orizontala : y = 0
y = mx +n , ec. asimpt. oblice
m = lim x-> -+inf (f(x)/x) = 0 (grad numarator < grad numitor)
n = lim x-> -+inf (f(x) -mx) =
lim x-> -+inf (f(x)) = 0
Nu are asimpt. oblice
