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Can anyone help me please? Don't know where to start since the numerator looks complicated. Thanks!

Can anyone help me please? Don't know where to start since the numerator looks complicated. Thanks!

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marked as duplicate by Ethan Bolker, Daniel Fischer real-analysis Jun 13 '17 at 12:48

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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In the case $x=y$ there is nothing to prove. Thus suppose $x>y$ wlog. Then the mean value theorem shows $$ \frac{e^x-e^y}{x-y}=e^\xi\leq e $$ for some $\xi\in(0,1)$. This proves the claim.

Your task is now to check whether the mean value theorem was applicable at all.

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