# Evaluate the limit by using epsilon-M approach. [duplicate]

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

## marked as duplicate by Ethan Bolker, Daniel Fischer real-analysis StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Jun 13 '17 at 12:48

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.