So the numerator x minus a and we're dividing by x minus a, those cancel out, and then these two cancel out, and we're left with this thing over here is equal to the limit as x approaches a of, in the numerator we have f of x minus f of a, and in the denominator we have g of x minus g of a. I think you see where this is going. What is f of a equal to? Well, we assumed f of a is equal to zero. That's why we're using L'Hospital's Rule from the get-go. f of a is equal to zero, g of a is equal to zero. f of a is equal to zero, g of a is equal to zero, and this simplifies to the limit as x approaches a of f prime of x, sorry, of f of x, we've got to be careful, of f of x over g of x. So we just showed that if f of a equals zero, g of a equals zero, and these two derivatives exist, then the derivatives evaluated a over each other are going to be equal to the limit as x approaches a of f of x over g of x, or the limit as x approaches a of f of x over g of x is going to be equal to f prime of a over g prime of a.