Lim x approaches 1 [(e^x - e) / (x - 1)]

Question

Can someone please algebraicly explain to me why the answer to this is "e"? Without L'hospital please. 

 

Thank you.

Answer 1

x=1 is the restriction here.  So evaluate the limit near x=1

 

Evaluate at x=0.9999    and    x = 1.0001

 

If the limits are the same, then there is a constant limit.  If the limits are not the same, then the limit does not exist.

 

 

Answer 2

Congratulations on correctly parenthesizing this.  Most student fail in this respect.

 

Direct substitution yields 0/0, so use l'HÔPITAL'S rule.

 

lim (x→1)  (e^x)/1 = e.