lim(n → ∞ ) (2^(n+1)+3^(n+1))/(2^(n)+3^(n))
lim(x → 2 ) (tgx-tg2) / (sin ln(x-1))
Делим на 3^(n+1)
=lim_(n→∞)((2/3)^(n+1) +1)/((1/3)*(2/3)^(n)+(1/3))=(0+1)/(0+(1/3)=1/(1/3)=3
(2/3)^(n+1)→0 и (2/3)^(n)→0 ( показательная функция c основанием (2/3) убывает)
2)
x→2 ⇒ x-2 → 0
Замена
x-2=t
x=(t+2)
=lim_(t→0)(tg(t+2)-tg2)/sin(ln(t+1))= применяем формулу разности тангенсов=
=lim_(t→0)sin(t+2-2)/(cos(t+2)*cost)*sin(ln(t+1)))=
=(1/cos2)*lim_(t→0)sint/sin(ln(t+1))=1/cos2
так как
(lim_(t→0)(sint/t))*(lim_(t→0)(t/sin(ln(t+1))=
=1*(lim_(t→0)(t/sin(ln(t+1))=(lim_(t→0)(t/ln(t+1))*(lim_(t→0)ln(t+1)/sin(ln(t+1))=1*1
(lim_(t→0) cos(t+2)=cos2
(lim_(t→0)cost)=cos0=1