If f(x) is derivable at x = a, then lim_"x rarr a"(xf(a) - af(x))/(x - a) is ??

1 Answer
Aug 14, 2017

f(a)-a f'(a)

Explanation:

(xf(a) - af(x))/(x - a) = (a x)((f(a)/a- f(x)/x)/(x-a)) Calling now g(x) = f(x)/x and x = a + h we have

(xf(a) - af(x))/(x - a)=(a (a+h))(g(a)-g(a+h))/h = -(a(a+h))(g(a+h)-g(a))/h

and then

lim_(x->a)(xf(a) - af(x))/(x - a) = -lim_(h->0)(a(a+h))(g(a+h)-g(a))/h =

= -a^2g'(a) = - a^2((f'(a))/a-f(a)/a^2) = f(a)-a f'(a)