If f(x) is differentiable at x=a where a is not equal to 0,

Question

Find lim [[f(x)]^2-[f(a)]^2] / [x^2-a^2]
as x approaches a

Answer 1

lim x->a [[f(x)]^2-[f(a)]^2] / [x^2-a^2]
=lim x->a [(f(x)+f(a))(f(x)-f(a)]/[(x+a)(x-a)]
=lim x->a [ (f(x)+f(a))/(x+a) * (f(x)-f(a)]/(x-a) ]

The first term evaluates to
2f(a)/2a
=f(a)/a

The second term is precisely the definition of f'(a) as h=(x-a) -> 0

Therefore the limit is
(f(a)/a)*f'(a)