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 Homework Statement

Let f and g be derivable functions and let a be a real number such that
##f(a)=g(a)=0 ##
##g'(a) ≠ 0 ##
Justify that
##\frac{f'(a)}{g'(a)} ## = ##\lim_{x\to a}\frac{f(x)}{g(x)}##
You may only use the definition of the derivative and boundary rules.
 Homework Equations
 ##\lim_{h\to 0}\frac{f(x+h)f(x)}{h}##
My attempt:
##\frac{f'(a)}{g'(a)} ## =
##\lim_{h\to 0}\frac{f(a+h)f(a)}{h}\cdot\frac{h}{g(a+h)g(a)}##
= ##\lim_{h\to 0}\frac{f(a+h)f(a)}{g(a+h)g(a)}##
I don't think I am doing this right. I don't even understand how I am supposed to use the boundary rules. I really appreciate some help!
##\frac{f'(a)}{g'(a)} ## =
##\lim_{h\to 0}\frac{f(a+h)f(a)}{h}\cdot\frac{h}{g(a+h)g(a)}##
= ##\lim_{h\to 0}\frac{f(a+h)f(a)}{g(a+h)g(a)}##
I don't think I am doing this right. I don't even understand how I am supposed to use the boundary rules. I really appreciate some help!