高数简单极限问题

Question

比如一个数列无限接近于0，则极限为0
那我为什么不能说极限是0.1或者0.01之类的。
刚刚学，有点懵懂。谢谢大家

Answer 1

(1) lim(x→1)(x^2-2x+1)/(x^2-1)=lim(x→1)(x-1)^2/[(x-1)(x+1)]=lim(x→1)(x-1)/(x+1)=0
(2) lim(x→4)(x^2-6x+8)/(x^2-5x+4)=lim(x→4)(x-2)(x-4)/[(x-1)(x-4)]lim(x→4)(x-2)/(x-1)=2/3
(3) 原式=lim(x→2)(x+2)/[(x-2)(x+2)]=∞
(4) 原式=lim(n→∞)1/2[1-1/3+1/3-1/5+……+1/(2n-1)-1/(2n+1)]=lim(n→∞)1/2[1-1/(2n+1)]=1/2
(5) 原式=lim(x→0)x^2[1+√(1+x^2)]/(-x^2)=lim(x→0)[1+√(1+x^2)]=2
(6) 原式=lim(n→∞)3/[√(x^2+1)+√(x^2-2)]=0
(7) ∵lim(x→0)x^2=0 |sin(1/x)|<=1 ∴lim(x→0)x^2|sin(1/x)=0
(8) ∵lim(x→∞)1/x=0 |arctan x|<π/2 ∴lim(x→∞)arctan x/x=0