Evaluating differentiable function in terms of limit

Question

How do you approach a question like this when you don't know the function? The answer is in terms of p and f'(x), but how do you write this limit in terms of the limit that denotes f'(x)?

Answer 1

$$ \lim_{h \to 0} \frac{f(a+ph)-f(a-ph)}{h}\\ = \lim_{h \to 0} \frac{f(a+h)-f(a-h)}{h/p} \\= p \cdot \lim_{h \to 0} \frac{f(a+h)-f(a-h)}{h} \\ = p \cdot \lim_{h \to 0} \frac{f(a+h) -f(a) +f(a)-f(a-h)}{h} \\ = p \cdot (\lim_{h \to 0} \frac{f(a+h) -f(a)}{h} + \lim_{h \to 0} \frac{f(a-h)-f(a)}{-h}) \\= p \cdot (f'(a) + f'(a)) = 2pf'(a) $$