f(x) = xlnx / [(x+1)(x-1)] (x≠1)
(x→1) lim f(x) = 0.5
(x→0+) lim f(x) = 0
(x→+∞) lim f(x) = 0
Plot the curve for f(x),and we can see:
when x→1,f(x) → 0.5
when x∈(0,1):lnx1,x-11:lnx>0,x+1>0,x-1>0
0 < f(x) < 0.5,f(x) is decreasing
Note:There is no definition for f(x) at x=1,which is a point of discontinuity of f(x).There is no value for f(x) at this point.The value of 0.5 is just the limit of f(x) as x approaches 1.The values of f(x) for any x in its domain [x∈(0,+∞),x≠1] are less than 0.5,which can be seen from its curve.