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Proof of special case of l'Hôpital's rule
What I want to go over in this video is a special case of L'Hopital's Rule. And it's a more constrained version of the general case we've been looking at. But it's still very powerful and very applicable. And the reason why we're going to go over this special case is because its proof is fairly straightforward and will give you an intuition for why L'Hopital's Rule works at all. So the special case of L'Hopital's Rule is a situation where f of a is equal to 0. f prime of a exists. g of a is equal to 0. g prime of a exists. If these constraints are met, then 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. So it's very similar to the general case. It's little bit more constrained. We're assuming that f prime of a exists. We're not just taking the limit now. We're assuming f prime of a and g prime of a actually exist. But notice if we substitute a right over here we get 0/0. But that if the derivatives exist we could just evaluate the derivatives at a, and then we get the limit. So this is very close to the general case of L'Hopital's Rule. Now let's actually prove it. And to prove it, we're going to start with the right hand and then show that if we use the definition of derivatives, we get the left hand right over here. So let me do that. So I'll do it right over here. So f prime of a is equal to what, by the definition of derivatives? Well, we could view that as the limit as x approaches a of f of x minus f of a over x minus a. So this is literally just a slope between two points. So like, if you have your function f of x like this, this is the point a, f of a right over here. This right over here is the point x, f of x. This expression right over here is the slope between these two points. The change in our y value is f of x minus f of a. The change in our f value is x minus a. So this expression is just the slope of this line. And we're just taking the-- let me actually do that in a different color-- the line that connects these two points, that's the slope of it. I'll do that in white. The slope of the line that connects those two points. And we're taking the limit as x gets closer and closer and closer to a. So this is just another way of writing the definition of the derivative. So that's fine. Let's do the same thing for g prime of a. So f prime of a over g prime of a, is going to be this business which is in orange, f prime of a over g prime of a. Which we can write as the limit as x approaches a of g of x minus g of a over x minus a. Well, in the numerator, we're taking the limit as x approaches a, and in the denominator, we're taking the limit as x approaches a. So we can just rewrite this. This we can rewrite as the limit as x approaches a of all this business in orange. f of x minus f of a, over x minus a, over all the business in green. g of x minus g of a, all of that over x minus a. Now, to simplify this, we can multiply the numerator and the denominator by x minus a to get rid of these x minus a's. So let's do that. Let's multiply by x minus a over x minus a. 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. And I think you see where this is going. What is f(a) equal to? Well, we assumed f of a is equal to 0. That's why we're using L'Hopital's Rule from the get go. f of a is equal to 0, g of a is equal to 0. f of a is equal to 0. g of a is equal to 0. 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 0, g of a equals 0, and these two derivatives exist, then the derivatives evaluated at 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. So fairly straightforward proof for the special case-- the special case, not the more general case-- of L'Hopital's Rule.
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