Its not difficult if you visualise it graphically.  
Search google for desmos graphing calculator and plot the graph for this function y= mod(x)/x.
You will see why left hand limit for thins function is -1 at 0.
 
Aletrnate solution 2
by definition mod(x) = x if x> 0 and
                                = -x if x 
Hence for x
or limit x->0- = -1.
 
Alternate solution2
let x =0-h
as h->0 x->0-
hence the given limit can we evalauted as
limit h->0  mod (h)/(-h) = -1.
 
Its as simple as that
 
 
						

							
Posting again as the there is bug in website which does not properly dispay inequalities.
 
Its not difficult if you visualise it graphically.  
Search google for desmos graphing calculator and plot the graph for this function y= mod(x)/x.
You will see why left hand limit for this function is -1 at 0.
 
Alternate solution 1
by definition mod(x) = x if x is greater than 0
                                = -x if x is less than 0
Hence for x less than 0
mod(x)/x = -x/x = -1.
or limit x->0- = -1.
 
Alternate solution2
let x =0-h
as h->0 x->0-
hence the given limit can we evalauted as
limit h->0  mod (h)/(-h) = -1.
 
Its as simple as that