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

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1 Answer
Cesareo R.
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)#

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