若f(x)在x=a处可导,求lim x->a xf(a)-af(x)/x-a

问题描述:

若f(x)在x=a处可导,求lim x->a xf(a)-af(x)/x-a

[xf(a)-af(x)]/(x-a)=[(x-a)f(a)+a(f(a)-f(x))]/(x-a)=f(a)+a[f(a)-f(x)]/(x-a),
==> lim x->a {[xf(a)-af(x)]/(x-a)}
=f(a)+lim x->a {a[f(a)-f(x)]/(x-a)}
=f(a)-a lim x->a {[f(x)-f(a)]/(x-a)}
=f(a)-af'(a).