The correct option is
D does not exist
Let,
L=limx→elnx−1|x−e|.Now the right hand limit i.e.
RHL =limx→e+lnx−1x−e 00 form
Now L'Hospital's rule we get,
or, RHL =limx→e+1x1=1e.......(1).
Again, the right hand limit i.e.
LHL =limx→e−lnx−1−x+e 00 form
Now L'Hospital's rule we get,
or, LHL =limx→e−1x−1=−1e.......(2).
Since RHL≠LHL, so the given limit doesn't exit.