Question

高数第五大题第一问 是否具有可导性

Answer 1

f(x) = (1-cosx)/√x , x>0
= x , x≤0
f(0) = f(0-) = 0
f(0+) = lim(x->0+) (1-cosx)/√x (0/0)
= lim(x->0+) sinx/[1/(2√x)]
= lim(x->0+) 2√x.sinx
=0
f(0) =f(0-)=f(0+) =0
f(0) 连续
f'(0-) =1
f'(0+) = lim(h->0+) [f(h) - f(0) ]/h
=lim(h->0) [(1-cosh)/√h ]/h
=lim(h->0) (1-cosh)/h^(3/2) (0/0)
=lim(h->0) sinh/[(3/2)h^(1/2)] (0/0)
=(2/3)lim(h->0) cosh/[(1/2)h^(-1/2)]
=(1/3)lim(h->0) √h.cosh
=0≠ f'(0-)
f'(0) 不存在