We must calculate the derivative #f'(x)=(u'v-uv')/v^2#

#u=x##=>##u'=1#

#v=x^2+1##=>##v'=2x#

#:. f'(x)=(1*(x^2+1)-x*2x)/(x^2+1)^2=(x^2+1-2x^2)/(x^2+1)^2#

#=(1-x^2)/(x^2+1)^2=((1+x)(1-x))/(x^2+1)^2#

#f'(x)=0# when #x=1# and #x=-1#

We do a sign chart for #f'(x)#

#color(white)(aaaaa)##x##color(white)(aaaaa)##-oo##color(white)(aaaaa)##-1##color(white)(aaaaa)##1##color(white)(aaaaa)##+oo#

#color(white)(aaa)##f'(x)##color(white)(aaaaaaaaa)##-##color(white)(aaaaa)##+##color(white)(aaaa)##-#

#color(white)(aaa)##f(x)##color(white)(aaaaaaaaaa)##darr##color(white)(aaaaa)##uarr##color(white)(aaaa)##darr#

The limits when #f(x)# is increasing is # x in [-1, 1] #

The limits when #f(x)# is decreasing is # x in ] -oo,-1 [ uu ] 1, oo[ #

graph{x/(x^2+1) [-3.077, 3.08, -1.538, 1.54]}