Задача 31205 ...

bolanebyla

2018-11-22 02:04:25

Найти пределы:

lim (x - sqrt(x² - x + 1))
x -> +∞.

sova

2018-11-22 13:29:58

★

Имеем неопределенность ( ∞ - ∞).
Умножаем и делим на (х+sqrt(x^2-x+1))

lim_(x → + ∞)(x-sqrt(x^2-x+1) =

=lim_(x → + ∞)(x-sqrt(x^2-x+1))* (x+sqrt(x^2-x+1))/(x+sqrt(x^2-x+1) =

=lim_(x → + ∞)(x^2-(sqrt(x^2-x+1))^2)/(x+sqrt(x^2-x+1)) =

=lim_(x → + ∞)(x^2-x^2+x-1)/(x+sqrt(x^2-x+1)) =

=lim_(x → + ∞)(+x-1)/(x+sqrt(x^2-x+1)) =

Делим и числитель и знаменатель на х:

=lim_(x → + ∞)(1-(1/х))/(1+sqrt((x^2-x+1)/x^2)) =

=lim_(x → + ∞)(1-(1/х))/(1+sqrt(1 -(1/x)+(/x^2)) )=1/(1+1)=1/2