My interpretation of the question is: what is the limit as x approaches zero of (1/2)ln(e^2x)+4-ln((5-x)/x)-ln(x+1).
Two log terms approach zero: ln(e^2x) becomes ln(1)=0 and ln(x+1) becomes ln(1)=0; but the central log term -ln((5-x)/x) approaches -ln(5)+ln(0) approaches minus infinity because ln(0) is minus infinity.
Interpretation of the central log term is crucial to the answer, because all the other terms are either finite values, or approach finite values. If the central log term is to read ln(5-(x/x)), the result is different, because x/x is 1 as x approaches zero. Then the expression becomes: 0+4-ln(4)-0=4-ln(4)=2.6137 approx.