What interval is #F(x) = (x^2)/(x^2+3)# increasing, decreasing?

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Answer 1 · 2016-12-07T12:42:00

#F(x)# is increasing for #x in (0, +oo)# and decreasing for #x in (-oo, 0)#

#F(x) = x^2/(x^2+3)#

#= 1/(1+3/x^2)#

Hence: #Lim_"x->+oo" F(x) = 1# and #Lim_"x->-oo" F(x) = 1#

Also notice, #F(0) = 0# which is an absolute minimum for #F(x)#

Therefore:
#F(x)# decreases from 1 for #x<0# and increases to 1 for #x>0#

This can be seen from the graph of #F(x)# below:

graph{x^2/(x^2+3) [-7.025, 7.02, -3.51, 3.51]}