I am somewhat confused by the proof of L'Hôpital's Rule in Pugh's "Real Mathematical Analysis." (See Attachments, Theorem 6). I follow every bit of the proof save one choice and its implication. That is, why choose t based on(adsbygoogle = window.adsbygoogle || []).push({});

##\displaystyle |f(t)| + |g(t)| < \frac{g(x)^2\epsilon}{4(|f(x)|+|g(x)|)}##

##\displaystyle |g(t)| < \frac{g(x)}{2}##,

and how does it follow that

##\displaystyle \left |\frac{g(x)f(t) - f(x)g(t)}{g(x)(g(x)-g(t))} \right | < \frac{\epsilon}{2} ##?

I understand that t need be chosen to be much closer to b than x, but I have yet to be able to convince myself why this meets that goal. Thanks for the help in advance!

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# Proof of L'Hôpital's Rule

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