Question

请问这道题x=0时的左极限和右极限怎么求，为什么符号不同？x=1和x=2时的极限为什么是无穷？

Answer 1

x->1时，分子趋于-sin1而分母趋于0，当然趋于无穷
当x->2时，分子~(x-2)，分母~-(x-2)^2，当然也趋于无穷大，因为分母是高阶无穷小

Answer 2

f(x)= |x|sin(x-2)/[x(x-1)(x-2)^2]

x(x-1)(x-2)^2=0

x=0 or 1 or 2

lim(x->0+) f(x)

=lim(x->0+) |x|sin(x-2)/[x(x-1)(x-2)^2]

=lim(x->0+) xsin(x-2)/[x(x-1)(x-2)^2]

=lim(x->0+) sin(x-2)/[(x-1)(x-2)^2]

=sin(-2)/[(0-1)(0-2)^2]

=sin2/4

lim(x->0-) f(x)

=lim(x->0-) |x|sin(x-2)/[x(x-1)(x-2)^2]

=lim(x->0-) -xsin(x-2)/[x(x-1)(x-2)^2]

=lim(x->0-) -sin(x-2)/[(x-1)(x-2)^2]

=-sin(-2)/[(0-1)(0-2)^2]

=-sin2/4

ans: A

f(x)= |x|sin(x-2)/[x(x-1)(x-2)^2] 在（-1，0）内有界

追问

请问为什么x=1和x=2时的左右极限都是无穷？

追答

请问为什么x=1和x=2时的左右极限都是无穷？是