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Math Help - Limit proof required for special case of chain rule

  1. #1
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    Limit proof required for special case of chain rule

    I am trying to figure out how to prove the equality I circled below in red. I have figured out how to prove the text in blue but don't know how to use that to prove the equality I circled in red.
    Below I will post the givens I'm trying to use and my guess of how to prove it.
    P.S. I understand this only proves the chain rule in the special case where \Delta u \neq 0 . This is from Stewart's Calculus and he does mention that this is not a full proof but I'm very curious how to prove this special case anyway.




    I'm not sure if I'm using the right givens below.
    In their blue form it looks like I will be able to use the transitive property of implication  (a \rightarrow b  \wedge b \rightarrow c)  \rightarrow (a \rightarrow c) if I can always let the \epsilon from the 1st given equal the \delta from the second line.
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    Re: Limit proof required for special case of chain rule

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  3. #3
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    Re: Limit proof required for special case of chain rule

    I'm not looking for a way to prove the product of the limits is the limit of the product. I'm looking for a way to prove the two expressions I circled in red are equal, i.e. \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x} = \lim_{\Delta u\to 0}\frac{\Delta y}{\Delta x}
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