f(x)在x=2处可导,且limx趋近于2f(x)/(x^2-4)=2,则f(2)=?,f(2的导数等于多

Question

f(x)在x=2处可导,且lim x趋近于2f(x)/(x^2-4)=2,则f(2)=?,f(2
的导数等于多

Answer 1

x趋近于2不好写,我就简略了,下面的lim就代表lim x趋近于2
lim (f(x)/(x^2-4))=2
lim(f(x)/(x+2)(x-2))=2
lim(f(x)/(x-2))/lim(x+2)=2
lim(f(x)/(x-2))=8
f(x)在x=2处可导
f'（2）=lim(（f（x）-f（2）)/(x-2))
=lim(（f（x）-f（2）)f(x)/f(x)(x-2))
=8lim(（f（x）-f（2）)/f(x))
故lim(（f（x）-f（2）)/f(x))有意义,又lim（f（x）-f（2）)=0
故limf（x）=0,即f（2）=0
f'（2）=lim(（f（x）-f（2）)/(x-2))
=lim(f（x）/(x-2))=8