Question

求极限lim(x→∞)√(x^2+x+1)/(x-1)

Answer 1

lim(x->∞)√(x^2+x+1)/(x-1)

分子分母同时除x

=lim(x->∞)√(1+1/x+1/x^2)/(1-1/x)

=√(1+0+0)/(1-0)

=1

Answer 2

lim(x→+∞)√(x^2+x+1)/(x-1) = lim(x→+∞)√(1+1/x+/x^2)/(1-1/x) = 1;
lim(x→-∞)√(x^2+x+1)/(x-1) ， 令 t = - x
= lim(t→+∞)√(t^2-t+1)/(-t-1) = lim(t→+∞)√(1-1/t+1/t^2)/(-1-1/t) = -1
故 lim(x→∞)√(x^2+x+1)/(x-1) 不存在。