Question

讨论函数f(x)={x^2sin1/x and 0在x=0处的连续性与可导性

Answer 1

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

=lim(x-->0)x*sin(1/x)/(1/x)

=0

即lim(x->0)=f(0)

所以f(x)在x=0处连续.

lim(Δx-->0)[f(0+Δx)-f(0)]/Δx

lim(Δx-->0)Δxsin(1/Δx)

=lim(Δx-->0)sin(1/Δx)/(1/Δx)

=1

f(x)在x=0处可导.