#lim x → ∞ (-x-1)/((x-1)(x-sqrt(x^2+x+1))#
#= lim x → ∞ (-x-1)/((x-1)(x-sqrt((x+ 1/2)^2+3/4))#
#y = x + 1/2 qquad x = y - 1/2#
#= lim y → ∞ (-y-1/2)/((y-3/2)(y - 1/2-sqrt(y^2+3/4))#
#= lim y → ∞ (-1-1/(2y))/((y-3/2)(1 - 1/(2y)-sqrt(1+3/(4y^2)))#
By binomial expansion:
#sqrt(1+3/(4y^2) )= 1 + (1/2)3/(4y^2) + bbbO(1/y^6) #
#= lim y → ∞ (-1-1/(2y))/((y-3/2)(1 - 1/(2y)-1 - (3/(8y^2)) ))#
#= lim y → ∞ (1+1/(2y))/((y-3/2)( 1/(2y) + 3/(8y^2) ))#
#= lim y → ∞ (1+cancel(1/(2y)))/(1/2 + cancel(3/(8y)) - cancel(3/(4y)) - cancel(9/(16y^2)))#
#= 2#