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# Proof of special case of l'Hôpital's rule

## Video transcript

what I want to go over this in this video is a special case of lapa tiles lapa tiles 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 it's proof is fairly straightforward and will give you an intuition for why lapa Tal 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 exists G of a G of a is equal to 0 0 G prime of a exists if these constraints are met then the limit the limit as X approaches a of f of X f of X / / G of X over G of X is going to be equal to F prime of a is going to be equal to F prime of a over G prime of a over G prime of a so it's very similar to the general case it's a 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 over 0 but then if the derivatives exist we can 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 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 over X minus a so this is literally just the slope between two points so like if you have your function f of X like this this is the point this is the point a comma F of a right over here this right over here is the point X comma f of X this expression right over here is the slope between these two points the change in the change in our y-value is f of X minus f of a the change in our x 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 this is 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 in the denominator taking the limit as X approaches a so we can just rewrite this this we could rewrite as a limit 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 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 so let's do that let's multiply by X minus a X minus a over X minus a so in 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 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 a lot tal's rule to get me from the get-go F of a is equal to zero G of a is equal to 0 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 F prime of X sorry of f of X we got to be careful of f of X over 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 of 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